Responding to Ed George About Mathematics

[Deleted content. - WJM] ET

WJM

As long as I have editorial control over my threads, statements or insinuations about the motives, intelligence or character of other people involved in the conversation will be deleted.

Fair enough. But could you please apply it to comment 278 as well. It links to an article that talks about the motivations of commenters. Ed George

As long as I have editorial control over my threads, statements or insinuations about the motives, intelligence or character of other people involved in the conversation will be deleted. - WJM That said, did everyone just abandon the discussion over at the three-knockdown-proofs-of-the-immateriality-of-mind thread? I thought it was just getting really interesting. William J Murray

[Deleted content. - WJM] kairosfocus

H, there is a substantial demonstration on the table, which shows that in any possible world, a substantial core of facts of structure and quantity will be embedded in the framework. This suffices to demonstrate that there is a considerable body of substance of structure and quantity that is discovered, not a mere property of our invention. That settles the substantial matter. [Deleted content - WJM] KF kairosfocus

EG, again, regrettably, you have moved to personalities and projections when a clear matter of demonstrable facts is on the table. The implication is that the matter on the merits can be taken as settled; as were the substantial matter in error this would have been long since jumped on. Where, we here deal with matters that are logically, structurally and quantitatively demonstrable. That demonstration having been provided and there being no credible reason to infer that it is in error, the substantial matter is effectively settled. In that light, where clear errors of argument -- aka fallacies -- have come up in ways that would side-track the substantial issue, we can properly regard them as irrelevant to the substantial core of the matter. More broadly, it is an index of where we are as a civilisation through the corrosive impact of relativist thought that not even a logical-mathematical demonstration will be allowed to stand on its merits. That is a warning-sign. KF kairosfocus

Deleted - WJM Ed George

kf write, "The refusal to face same settles the matter," At the risk of repeating myself, baloney! :-) [Deleted content - WJM] hazel

EG, there is a substantial issue on the table as the core of the matter. [Deleted - WJM] kairosfocus

Deleted - WJM Ed George

EG, again, you set up and knocked over a strawman caricature. The pivotal issue is that in doing Mathematics we study something that is demonstrably embedded in the world, the substance of structure and quantity. Where, I can freely say "demonstrably" as I actually took time to demonstrate it. Thus, you show yourself unresponsive to objective warranted conclusions, evidently because they do not sit well with your opinions. You have then resorted to projecting to me a demand to impose my opinions. [Deleted content - WJM] A suggestion: if I am wrong on the merits of fact and logic above, all that would be required is to show why the steps of reasoning fail. Namely, kindly explain why any distinct world W will not have a partition A vs ~A: ___ and/or how such a partition does not entail duality, unity and nullity: _____, so also why the von Neumann construction fails: ____; thence, why the naturals do not lead to the integers, rationals, reals, complex numbers and space, thus linked properties, entities, relationships etc. ________ . Failing that, it is pretty clear who has had the better of the case on the merits [Deleted - WJM]KF kairosfocus

[Deleted content - WJM] Ed George

KF

EG, you left off the part about accountability before mathematical facts,...

I left it out because it is not relevant to my original opinion. Ed George

H, did you observe that I responded to a particular and very specific rhetorical move of fallacious character? I suggest that you take time to observe why I spoke correctively. Otherwise, you will fall to the he hit back first fallacy. Except, I was not throwing a mere rhetorical punch but analysing in brief why a particular argument tactic used above is fallacious. Part of duty to truth and right reason is to correct fallacies. KF PS: Let me add, that the issue of selective hyperskepticism is the demand for an inconsistently high standard of warrant for claim A when, had it been another more welcome comparable claim B, no such criterion would have been put up. The result of such a fallacy is that one exerts a double-standard of warrant, becoming unresponsive to well warranted claims. Notice, above, I laid out in summary just how any possible world will have embedded in it a considerable body of structure and quantity. This, being an aspect of the logic of being and thus antecedent to our particular exploration and study of this aspect of reality. kairosfocus

EG, you left off the part about accountability before mathematical facts, many of which are embedded in the structure of any possible world, as I have taken time to highlight again and again. This is a good part of why I used a dual-mode definitional framework: Mathematics is the (study of the) logic of structure and quantity. There is substance embedded in any possible world, there is further substance embedded in the particularities of a given world -- hence discussion of abstract, logic model worlds -- and such will be antecedent to our particular cultural or research tradition. So, we distinguish substance and study. Truncating Mathematics to focus on the tradition fails to recognise that that tradition works with and is decisively shaped by that antecedent substance. We explore the substance and discover truths that are there, locked into the logic of structure and quantity. We can do that in Greek, Sanskrit, German, Russian, Chinese, Arabic, Spanish, French or English etc, using diverse possible frameworks. But just like the emeralds being there in the ground over in Columbia, the exploration is discovering something that was there independent of the exploration. KF kairosfocus

[Deleted content - WJM] hazel

KF

ET, I will grant them that mathematical models and axiomatic systems constitutive of abstract logic model worlds are in large part constructions of human traditions.

Unless I am reading this wrong, this is essentially the opinion I expressed that started this whole thing. Ed George

ET, I will grant them that mathematical models and axiomatic systems constitutive of abstract logic model worlds are in large part constructions of human traditions. However, those traditions -- if they are to work (starting with coherence) have to be accountable to a large body of Mathematical facts that obtain in any possible world. Many of those facts are outright self-evident and/or are connected to the logic of being. Thus, every possible world has deeply embedded in it the logic of structure and quantity. Reality cannot but be pervaded by that logic. It is then obvious that reality is best explained as coming from (and as being expressive of) utterly rational mind. I suspect, this onward issue is the silent ghost that is driving the sort of resistance we observe on this issue . . . far beyond exchanges on this thread. KF kairosfocus

[Deleted content - WJM]? That is another facet of the underlying problem. I'll add that I am very aware of how entrenched schools of thought can resist evidence, I came of intellectual age in the days of the surge then collapse of Marxism, and saw how they operated until things fell apart internally, 1989 - 91. Von Mises' analysis from the 1920's proved correct and was manifestly correct, but it took 60 years for a degenerative research programme and its agenda to break. In this case, the clear evidence is that there is a lot of the logic -- rational principles -- of structure and quantity embedded in any possible world (and tied to the logic of being), antecedent to our coming along to explore it within whatever traditions. That is what needs to be faced squarely. KF kairosfocus

Hazel- I have provide the arguments and the evidence. You have provide nothing but your opinion- not even an argument. ET

ET: and it is your opinion that the universe was intelligently designed using mathematics. Different opinions. hazel

hazel:

Rather it is a conclusion based on a philosophical assumption that reasonable people disagree about.

That is your opinion. Srinivasa Ramanujan is evidence that math is discovered and not invented. [Deleted content - WJM] No one has presented any evidence that mathematics- of any kind- was invented. ET

It isn't my opinion, Ed. [Deleted content - WJM] ET

Hazel@256 and 257. Nicely said. Disagreement and discussion are enjoyable and often informative. Ed George

[Deleted content - WJM] hazel

Ed writes, "With the huge caveat that neither of us is a mathematician nor a cosmologist,...." Actually, I am a minor-league mathematician, and it appears kf is also. Perhaps we have read and thought about these things more than the average person, and have more experience solving problems with math. However, that gives us no special status in respect to the philosophy of math, I don’t think. Also, FWIW, I don’t think ET’s comment that “Our universe is such that math can provide accurate descriptions of the world because the universe was intelligently designed using mathematics.” is technically a circular argument. Rather it is a conclusion based on a philosophical assumption that reasonable people disagree about. In fact, both ET’s philosophical premise and the Platonism that kf invokes are possible explanations of perennial issues that are explained differently by other perspectives. We aren’t going to settle these issues on an internet forum populated by a miscellaneous, self-selected group of people. Likewise, as Ed says, we aren’t obligated to going around and around about it, either. And, another FWIW, I don’t exactly agree with ED about the nature of math, or wouldn’t say some things as he does. But so what? None of us have the corner on the truth here. hazel

[Deleted content - WJM] kairosfocus

ET

Our opponents have yet to post an argument.

Everybody here is merely presenting opinions. With the huge caveat that neither of us is a mathematician nor a cosmologist, your opinion is that mathematics is inherent in the universe and mine is that it is not. Neither opinion has been supported by any arguments compelling enough to convince the other. I can live with that. At the end of the day, it doesn't make any difference either way. Frankly, I was surprised that anyone would draft an OP to respond to one of my opinions. I should be flattered, but the subject simply doesn't interest me enough to delve too deeply into it. KF

The selectively hyperskeptical veto fails.

[Deleted content - WJM]I have provided my reasons as to why i don't think that mathematics is inherent in the universe. You don't agree with my rationale. That is fine. I won't lose any sleep over it. To paraphrase Hazel, if we haven't come to an agreement after 250 comments, then it is reasonable to conclude that this comment thread has run its course. KF

H, I for cause expect due responsiveness to substantially warranted conclusions as were just again shown in outline, as manifesting duty to truth and to right reason.

Expecting and receiving are two different things. Nobody here has any obligation to respond to anybody else. That is the nature of blogs. Ed George

Our universe is such that math can provide accurate descriptions of the world because the universe was intelligently designed using mathematics. Ed sed that was a circular argument but in typical fashion couldn't support his claim. ET

H, I for cause expect due responsiveness to substantially warranted conclusions as were just again shown in outline, as manifesting duty to truth and to right reason. For example, I believe I can fairly claim to have substantiated the force of the extended XXX definition (study of structure) of what Mathematics is, doffing hat to my old prof. Where, too, we who are UD contributors -- who have borne the brunt of not only web rhetoric and cyber stalking but in some cases on the ground stalking out to relatives at several degrees of remove -- are ever cognisant of the wider audience, especially the penumbra of attack sites. KF kairosfocus

PS: The above demonstrates that in any possible world a large domain of abstracta with quantitative and structural properties and relationships necessarily obtains. Indeed, obtains in ways that are ontologically effective as connected to the logic of being. The abstract and the concrete cannot bbe severed, they are inextricably entangled. Where, necessary involvement of an abstract, structural, quantitative domain in any possible world then gives teeth to the claim that such abstract entities and domains are real, not merely figments of fevered cultural imagination. Somewhere out there, the shade of Plato is laughing. PPS: Objections notwithstanding -- given what we just saw -- let us note SEP:

Platonism in the Philosophy of Mathematics First published Sat Jul 18, 2009; substantive revision Thu Jan 18, 2018 Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths [--> I would add, in material part] are therefore discovered, not invented . . .

kairosfocus

kf, you repeat what you have said before, some of which I agree with and some I don't. ET isn't specific about what false claims ED made, but it seems his issue is with ED, not me. However, what I don't get is the "chirp, chirp, chirp" part. Did you expect Ed and/or I to just keep on commenting ad infinitum when it seemed like everyone had said what they had to say? There is nothing "telling" about thinking a conversation has run its course. hazel

EG, [Deleted content - WJM] There has been just one serious definition of Mathematics on the table, one which distinguishes the largely reality-embedded substance from the culturally influenced study which is constrained by the necessity of facing that substance. Namely, the [study of the] logic of structure and quantity. In that context, it is manifest that for a distinct possible world W to be, there must be some distinguishing characteristic, A. So, we freely proceed: W = {A|~A}, which instantly establishes duality, unity and nullity. The contrasted entities are distinct unities, thus a duality and the partition betwixt is empty of members. This, with the simple expression of the von Neumann succession: {} --> 0 {0} --> 1 {0,1} --> 2 {0,1,2} --> 3 etc . . . shows that the natural counting numbers are necessarily present in any possible world, which instantly leads to many relationships and properties eg primes, evens, Pythagorean triplets, neighbouring successor primes such as 11 & 13, Pascal's triangle etc. Further, additive inverses give integers as x + (-x) = 0, so for any x we have -x. Ratios give rationals and power series give the reals, with the transfinites all the way out to the full surreal panoply beckoning. Taking reals and applying the i* operator, and we have the complex plane, with quaternions beyond etc if you please. The ijk system of orthogonal unit vectors gives a flat space in which Euclidean spatial properties obtain. In ANY possible world. And much more. Thus, we see structures and quantity deeply embedded in any possible world in ways that manifestly transcend our culturally influenced study thereof. [Deleted content - WJM] KF kairosfocus

[Deleted content - WJM] ET

What exactly is the issue, ET, and what true claims that are more than bald assertions has "your side" made. Can you summarize? hazel

Ed George:

Neither side has presented any arguments compelling enough to change the other’s view.

Our opponents have yet to post an argument. All you are capable of is making false claims and then "supporting" them with bald assertions. ET

KF

Key issue is still on the table, a walkaway says something.

What key issue? We disagree on the nature of mathematics. Neither side has presented any arguments compelling enough to change the other's view. We could keep going back and forth for another 245 comments with no resolution,. or we could simply agree to disagree. Ed George

No, I don't think so. And, to be clear, what key issue are you referring to, and who walking away? I was involved here, and the contretemps between ET and EG at the end was pretty insubstantial, so what exactly are you referring to? hazel

[Deleted content - WJM] kairosfocus

??? This thread has been dead for days, and ended with your pointing out to EG and ET that back-and-forth wouldn't help things, So why the "chirp, chirp, chirp"? What were you expecting??? hazel

[Deleted content - WJM] kairosfocus

[Deleted content - WJM] I was actually looking forward to a possible learning moment. I am not perfect and I am far from infallible. And, given what I have learned about evolution by reading what evolutionary biologists have to say, clearly I am capable of learning points of view contrary to my own. Alas, it was not to be. I will not bring it up again unless said learning moment does arise. You have the floor, sir.

I wonder what the diameter of that circular argument is.

From all appearances it is i- as in imaginary. ;) ET

ET & EG, the back-forth will not help things. Besides, EG just failed to attend to a relevant correction. Further, the rhetorical abuse of self-evident also needs a comment. Namely, something self evident will, on correctly understanding it, be seen as necessarily true; e.g. 3 + 2 = 5, or error exists or that once one is self aware, s/he cannot be mistaken about that fact, or the first principles of right reason, etc. This, on pain of patent absurdity on the attempted denial. EG's assertion of circularity a bit above is unwarranted and in fact false. As was shown yesterday. KF kairosfocus

[Deleted content - WJM] ET

ET

So Ed George ran away instead of supporting its claim that I posted a circular argument.

No, still here. I don’t see the need to support something that is self-evidently true. Ed George

[Deleted content - WJM] ET

EG, per your declarations you won't read this, so this is for record; so that others will be able to duly take note of the balance on merits. For one, the embedding of the substance of structure and quantity in the observed cosmos goes far beyond the naturals or the patterns of a spatially extended world. We have dozens of specific cases of fine tuning that sets up a world suitable for C-Chemistry, aqueous medium, cell-based life; cf here for a summary. Where that life uses digitally coded algorithmic information for its workhorse molecules, the proteins; a case of language. Language is of course one of the strongest signs of intelligence. On pondering best, empirically anchored explanation i/l/o alternatives, it is a very serious contender indeed that the observed cosmos was intelligently configured to host just such cell based life by a mathematically extremely sophisticated designer. Where, as I just noted to H, comparative difficulties and inference to best current explanation suffice to remove vicious circularity as questions are asked and alternatives are weighed. KF PS: Notice, the design inference is an inductive inference on signs. The reasoning about the embedding of structure and quantity in possible worlds is tied to implications of distinct identity such that particular worlds are possible or actual. kairosfocus

What I said in 230 is the same type of argument as the 3 scenarios presented in 232- with the exception of two scenarios don't make any sense at all. The car turned to the right because it was designed to turn to the right when you turned the steering wheel to the right. You turn to the right by turning to the right. In that circular argument course they would teach people how to make a case instead of just saying so. Your say-so is wouldn't make it past the playground. :D ET

ET@232, no, no and no. But ET@ 230 was a circular argument. If there was a circular argument university, your example would be taught in Circular Argument 101. :) Ed George

Wow. Is it a circular argument to say the sun appears to rise in the east because of the way the earth rotates in relation to the sun? or Our universe is such that math can provide accurate descriptions of the world because the universe just happened? or Our universe is such that math can provide accurate descriptions of the world because the universe was intelligently designed but it was so designed without using mathematics? ET

ET

Our universe is such that math can provide accurate descriptions of the world because the universe was intelligently designed using mathematics.

I wonder what the diameter of that circular argument is. :) Ed George

Our universe is such that math can provide accurate descriptions of the world because the universe was intelligently designed using mathematics. ET

H, I spoke regarding Wikipedia, for cause. I would suggest that the force of the substance of structure and quantity embedded in reality has shaped the study far more than some may freely acknowledge. Your comment "I’ve repeatedly acknowledged that math starts with abstractions based on our experience of the world, and that our universe is such that math can provide accurate descriptions of the world" in effect acknowledges the significance and impact of that substance. KF kairosfocus

I don't disagree with anything Wikipedia says.. I don't think, however, that I have been "compelled" to discuss pure mathematics from any "ideological bent", and I've repeatedly acknowledged that math starts with abstractions based on our experience of the world, and that our universe is such that math can provide accurate descriptions of the world. So what's your point? hazel

H, As you commonly appeal to pure mathematics, let me cite Wikipedia speaking against known ideological bent under the compelling force of the well known circumstances:

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but the pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and esthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900,[1] after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need of renewing the concept of mathematical rigor and rewriting all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics. Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications. It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.

Patently, axiomatisation sets up abstract logic model worlds exhibiting intelligible rational principles [= logic] of structure and quantity. Such then may exhibit necessary entities, which will be present in all possible worlds. In other cases, relevant entities may be present in our world, as a case in point, hence utility. Those are general, and have often been pointed out. However, let's highlight and comment more specifically:

Pure mathematics is the study of [--> study vs substance] mathematical concepts [--> structural and/or quantitative] independently of any application outside mathematics. These concepts may originate in real-world concerns [--> come from the substance of structure and quantity we encounter] , and the results obtained may later turn out to be useful for practical applications . . . . almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories [--> again, ties to mathematical "facts" of structure and quantity] . Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science . . . . the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics [--> the forced distinction fails, we have to reckon with the dual character of substance and study] . . .

KF kairosfocus

Yes, learning about the power of deductive reasoning, starting from accepted assumptions (axioms, undefined terms, definitions, etc.), and moving on to proving things that were not obvious, was an important part of my intellectual growth as a teenager. hazel

F/N: More and more I see the wisdom in my old Jesuit teachers insisting that Geometry opens the mind to a new world of thinking. KF kairosfocus

H, we define the label, circle and define the particular ratio of interest, circumference to diameter. We most certainly did not invent the circle as a spatial form or its structural and quantitative properties; though we idealise round objects we experience. Indeed, that we recognise imperfections of round objects implies that we contrast with an ideal that we perceive with our minds, an item from an abstract logic model possible world with quantitative and structural aspects. To see this, consider: can we arbitrarily redefine what a circle is and change its properties or abolish circles from existing, not least as contained disk and locus such that z^2 = x^2 + i*y^2 for some set |z| = r, where x and y are real numbers; or translations and/or reflections thereof? Patently, not. Circularity is a spatial property of a class of figures, a special case of ellipses and one of five famous conical sections. Pi is indeed a logical consequence of the structure we term a circle, ratio of distance around the circumference to the length of the longest chord. Which chord has additional properties: it passes through the centre, which bisects it, a triangle standing on either end and having third vertex at the circumference will necessarily be a right angle triangle with the right angle at the third vertex, and much more. KF PS: As virtual worlds are difficult, try the exercise of setting up a plane mirror strip and using parallax to fix virtual image points for a triangle or other object. Soon, you will see that a plane mirror specifies a virtual half-universe behind its surface. kairosfocus

From a purely mathematical point-of-view. pi is a logical consequence of the formal definition of a circle. hazel

Symbols we pick and use are arbitrary but relationship between let's say circle circumference and circle diameter is not, it's just somehow embedded in reality. Some people struggle with this simple concept Eugen

ET, often the symbols refer to or even represent and instantiate the facts in question. We have long since shown that the naturals and extensions to C are embedded in any world or in the case of continuum any world that at least conceptually has a space. KF kairosfocus

hazel:

Pure mathematical facts reside within the symbol system of words and notations we have devised.

Just saying it doesn't make it so. ET

H, the properties of triangles and circles under stated circumstances have very specific quantitative implications. This reflects the way structure and quantity are embedded in reality; as it is coherent. You composed a specific problem as a part of our study, but that problem rests in turn on the substance. KF kairosfocus

Here is a scenario to consider, and some thoughts. Let me first say that I am not a philosopher, and don't like playing one in public. But the question of the nature of math is one for which I have practical experience, and some reading background. I have written literally hundreds of pages of math curriculum. Here is a geometry problem that I could have written for a trig class. You don't really need to fully grasp the problem from the description, since it is the philosophy I'm going to discuss, but you might draw a picture if you want to follow along. An isosceles triangle ABC, with vertex at A, is inscribed in a circle. The sides of ABC are 10, and the base is 5. What is the measure of arc AB? Note that especially if I changed the sides to some larger numbers, such as legs of 4356 and a base of 2314, this problem most likely has never existed before in the history of humankind. To me, it seems clear that I invented this problem: in fact, saying I created it might be better. Also, the moment I thought of it, even before I formalized it by writing it up for a worksheet or figured out the answer, I was absolutely sure that one and only one correct answer existed, and that I could easily discover the answer. What can I possible mean that the answer existed before I found it. That is the question I want to think about. (See footnote below for the answer.) If I try to explain this in terms of a Platonic realm, I can't come up with anything reasonable. One possibility is that the Platonic realm has eternally contained every possible mathematical situation, whether someone has thought of it or not: my problem, the Game of Life, the 1 millionth digit of pi in base 17, the status of every point in the complex plane in respect to the Mandelbrot set, etc. For reasons I have explained before, this just doesn't seem like a feasible explanation. Another explanation which occurs to me, but hasn't been mentioned, is that the moment I thought of the problem, it became a part of the Platonic realm, and thus my answer becomes instantly true even before I figure it out. This doesn't seem reasonable. I'm now going to explore another explanation, with no assurance that this will be reasonable either, and with my disclaimer about playing philosopher in mind. Rather than consider ontology, which is what a Platonic realm is about, I would like to just consider epistemology: what we know and how we know it. Logic is inexorable: given some premises within a given mathematical system, assuming the presence of other various theorems, terms, and techniques already established by logic, a whole set of logical conclusions follow. For instance, in my problem, not only could we find arc AB we could find the area inside the circle that is outside the triangle as well. So, in some sense, the potential logical conclusions that could be known are implicitly present in the beginning situation. Just because we haven't articulated the logical steps to reach the answer doesn't mean we can't. Thus, when we say that the answer exists even though we don't know it yet, I think the word "exists" is part of the confusion. It is not an ontological existence that we are talking about. What we are talking about is an epistemological existence: the answer is knowable within the system, but is not yet known to us as an articulated fact. Pure mathematical facts reside within the symbol system of words and notations we have devised. The development of math involves articulating logical consequences of what we already know (including the steps to establish them). We move math facts from unknown to known, which is an epistemological change, but we don't change the fact that they exist within a symbol system: ontologically they remain elements of our logical system whether they are unknown or uknown. =========== Another disclaimer: I know that math can do a marvelous job describing various aspects of our physical world. I don't want to play philosopher on why that is true. I accept it as a fact. But this post has been about pure mathematics, and I want to leave it at that. -------------------------- Footnote: The answer is arc AB = 2 * arc cos (1/4), which I will leave as exercise for the reader. Another solution is arc AB = 180 - arc cos (7/8). This illustrates the point that there can be multiple logical paths to a fact, and that the fact can be expressed in different but logically equivalent ways. Furthermore, I notice that since angle A = arc cos (7/8), arc BC + angle A = 180: they are supplements. hazel

Here are reasons as to why I believe that mathematics (mathematical truth) is discovered not invented. *1. Numbers have properties that do not appear to have been invented. For example, there some unsolved conjectures about prime numbers that are hard to explain if we are the inventors. Namely if we are the inventors why has no one been able to prove (or disprove) that the set of twin primes is infinite? Or why do the Goldbach conjecture and Riemann Hypothesis continue to be unsolved? Wouldn’t the putative inventors of mathematics be able to resolve these problems? *2. The applicability of mathematics to the physical world. For example, earlier in this thread I pointed out that “One of the most significant discoveries in science was the discovery of the inverse square law (credited to Kepler for light) which is derived directly from the geometry of a sphere. The ISL applies to both electromagnetism and gravity, though the force constants for each vary.” https://www.thehighersidechatsplus.com/forums/media/inverse-square-law-and-wave-function.105/full?d=1503980290 Where would physics be without this discovery? And that’s only one example. *3. It appears that the human mind and brain are preadapted to do mathematics. What survival advantage would doing math and doing it accurately have for a highly evolved species of hunter-gatherer apes? *4. The universality of mathematics. SETI enthusiasts have suggested that we could use mathematics to communicate with ETI’s. For example, “In the 1985 science fiction novel Contact, Carl Sagan explored in some depth how a message might be constructed to allow communication with an alien civilization, using prime numbers as a starting point, followed by various universal principles and facts of mathematics and science.” https://en.wikipedia.org/wiki/Communication_with_extraterrestrial_intelligence How could mathematics be universal if it was invented by us? *5. Historically mathematics set the stage for the scientific revolution. Kepler and Galileo and Newton were all mathematicians who believed that at its root the universe was mathematical. In other words, they began with the assumption that the universe could be described mathematically. Does anyone have anything to add? john_a_designer

H, note the summary from Needham in 211. Notice how properties of circular motion show how complex numbers act as vectors. Similarly, note how the exponential function emerges as a characteristic function under the operation D*. Onward, a lot of geometry comes out, including a whole perspective rooted in translations and reflections. More coherence and intelligibility of rational principles of structure and quantity. KF kairosfocus

P.S. I'm having some new thoughts that clarify things a bit for myself: hope to post them sometime today. hazel

I don't think so. I have also discussed reasons why I don't acknowledge platonic reality. The fact that there are logical conclusions that must be true even if we never explicitly follow the steps that lead to them says something about the nature of logical implication, but not necessarily anything about ontology. Reminder: I am of two minds (or more) about all this, but I can't agree that, based on one of the things I said, without taking others into account, that I acknowledge platonic reality. The things I said about about what I can't accept about the idea of platonic reality are deal-breakers for me. YMMV. hazel

hazel: I agree and have discussed above that one of the features of math is that the things we discover are true whether we have discovered them or not. If you what you're saying is that implications in our abstract maths are true whether they have been perceived by us or not, this seems to be an acknowledgement of the platonic reality. If you ponder this statement in a meditative state, you may experience a koan. ;) mike1962

Ed George:

but is this a case of discovering the mathematics inherent in the universe, or finding applications for the mathematics we invented.

You have yet to make a case that we invented anything. ET

PS: Tristam Needham develops thoughts at book length, note responses: http://usf.usfca.edu/vca/ and raises profound issues. He highlights that Newton employed two distinct approaches to Fluxions (aka Calculus), a power series one AND a later geometric one, as opposed to algebraic formalisms that sometimes help [and seem to be "more rigorous"] but which tend to lock out the visual-spatial aspects of insight into structure and quantity. For example, he applies the i-operator approach, asking what happens when on a continuous basis velocity -- trajectory in an ideal space -- is at right angles to position vector? This is of course bringing to bear the i* operation, and allows us to recall that this v- is- ever- tangential- to- z . . . the complex vector . . . is a key defining aspect of the kinematics of circular motion. At (1,0) of course velocity is directly upward, at (0,1) it will be leftward, and at (-1,0) it will be downward. In between, we will ever be pointing v in the leftward (anticlockwise) tangent direction to z considered from the origin, the pivot of the rotation. We thus see that, naturally, i* is about rotation and here circular motion. We can of course readily apply the orthogonal components of z, x --> r cos wt and y --> r sin wt. The three power series expansions dovetail once we use i* y, giving the link to z = r e^iwt. He makes some historical observations:

far from being embraced, complex numbers [which appeared -- with suspicion -- in 1545] were initially greeted with suspicion, confusion, and even hostility . . . . The root cause of all this trouble seems to have been a psychological or philo-sophical block. How could one investigate these matters with enthusiasm or confi-dence when nobody felt they knew the answer to the question, “What is a complex number?” A satisfactory answer to this question was only found at the end of the eigh-teenth century z. Independently, and in rapid succession, Wessel, Argand, and Gauss all recognized that complex numbers could be given a simple, concrete, geometric interpretation as points (or vectors) in the plane: The mystical quantity a + i b should be viewed simply as the point in the xy-plane having Cartesian coordinates (a, b), or equivalently as the vector connecting the origin to that point. See [1]. When thought of in this way, the plane is denoted (C and is called the complex plane3.

Sounds familiar? He continues, applying the vector approach:

The sum A +B of two complex numbers is given by the parallelogram (1) rule of ordinary vector addition. The length of AB is the product of the lengths of A and B, and the 2 angle of AB is the sum of the angles of A and B [from the polar axis, ox].

Back on the worldviews front:

The publication of the geometric interpretation by Wessel and by Argand went all but unnoticed, but the reputation of Gauss (as great then as it is now) ensured wide dissemination and acceptance of complex numbers as points in the plane. Perhaps less important than the details of this new interpretation (at least initially) was the mere fact that there now existed some way of making sense of these numbers——that they were now legitimate objects of investigation. In any event, the ?oodgates of invention were about to open.

The name for this, is paradigm shift, where endorsement by a star of the field (yes, academia has long had a celebrity culture) triggered an acceptance cascade . . . shifting the overton window for Mathematics. But 200 years later, too much of education on the subject has not caught up. (And the discussion on quadratics and cubics is useful too.) Another key insight (I substitute wt for theta and vt for phi and use / for the angle notation that looks like a capital L bent into acute angle shape):

Let z denote a general point [= position vector relative to origin and polar axis] in C, and consider what happens to it— where it moves to-—when it is multiplied by a fixed complex number A = R/vt. According to [the vector product rule], the length of z is magnified by R, while the angle of z is increased by vt. Now imagine that this is done simultaneously to every point of the plane: Geometrically, multiplication [of the plane of z's] by a complex number A = R/vt is a rotation of the plane through angle vt, and an expansion of the plane (9) by factor R.

Thinking spatially is already opening up new vistas. To do so, he brings in Euler's e^iwt = cos wt + i sin wt. This can be seen as a vector sum on components and can be expressed as power series expansions. He goes on:

Instead of writing a general complex number as z = r / wt, we can now write z = r e^ i*wt. Concretely, this says that to reach z we must take the unit vector e^i*wt that points at z, then stretch it by the length of z.

Going on, and using D* as derivative operator d[]/dx:

Recall the basic fact that e^x is its own derivative: D*e^x = e^x. This is actually a defining property, that is, if D* f(x) = f(x), and f(0) = 1, then f(x) = e^x. [--> characteristic function under the operation D*] Similarly, if k is a real constant, then e^kx may be defined by the property D*f(x) = k f(x). To extend the action of the ordinary exponential function e^x from real values of x to imaginary ones, let us cling to this property by insisting that it remain true if k = i, so that D* e^it = i*e^it . (11) We have used the letter t instead of x because we will now think of the variable as being time. [--> we have been there all along, think of positions (x,y) and trajectories manifested in velocities etc]

We already can apply that i* means A/C rot through a right angle and we see that we have rate of change of position here as perpendicular to position vector at all times. That is, naturally, we are looking at circular, vector motion in the plane. Following Needham:

. . . velocity = V = iZ = position, rotated through a right angle. Since the initial position of the particle is Z(0) = e^0 = 1, its initial velocity is i [upwards], and so it is moving vertically upwards. A split second later the particle will have moved very slightly in this direction, and its new velocity will be at right angles to its new position vector. Continuing to construct the motion in this way, it is clear that the particle will travel round the unit circle.

In short, we need multiple perspectives, which mutually reinforce. And here we see how structures and quantities tied to space [which requires 2-d continuum] allow us to see how complex numbers understood vectorially have a very natural interpretation. One, embedded in the realities of space. And note how rates and accumulations of change across time in space -- trajectories -- are also deeply involved. kairosfocus

F/N2: This discussion brings in cross-perspectives on defining exponential functions (in effect what is the characteristic function of differentiation?) and on the import of vector based trajectories: https://books.google.ms/books?id=ogz5FjmiqlQC&lpg=PP1&pg=PA10&redir_esc=y&hl=en#v=onepage&q&f=false KF kairosfocus

F/N: A useful video explanation of the Euler expression: https://www.youtube.com/watch?time_continue=24&v=qpOj98VNJi4 KF kairosfocus

MG, very well noted. In addition, there is elegant simplicity that embeds the profound . . . and thus excites the aesthetic/ axiological wonder triggered by beauty. In electronics, I once sat in a workshop by a genius circuit designer, who used a classic fixed bias transistor ckt in subtle ways. A lesson. Euler's expression as I discussed last evening is similar. KF kairosfocus

WJM @ 161 "I think the question now becomes: what can we infer from the fact that mathematics is an inherent aspect of the universe and of our existence?" First off, let's review the obvious fact that A-mat is incompatible with an infinite, eternal (mathematical) platonic realm. Such a realm requires intelligence (perhaps among other traits) to access. Aspects of the realm require genius to perceive (which is why my own access is rather minimal :-) We see some highly sophisticated mathematics that undergirds our physical universe, in particular QM which was discovered by genius minds such as Planck, Bohr, Einstein, Schroedinger, Heisenberg, Dirac, Feynman, et al. As BA77 is wont to extol, QM is full of counterintuitive facts. Even the great Einstein did not want to accept quantum entanglement, but it is now an established fact. Continuing by analogy, there should not be an upper bound on the complexity of relationships and laws in the platonic realm (in fact, Godel's theorems imply this). Some of the relationships/laws will be beyond human reach and only be accessible to godlike intelligence. I suspect that subtle undiscovered laws governing our own universe belong to that class. math guy

MG@24, but is this a case of discovering the mathematics inherent in the universe, or finding applications for the mathematics we invented. In your examples I would suggest that it is the latter. Ed George

I totally agree, MG. I'm pretty sure I said somewhere previously in this thread that a lot of math has first been developed as pure math, and then found to have application to the real world. Complex numbers, which we have discussed a lot here, are one such topic. hazel

h @ 201 "And lots of math has been logically discovered that doesn’t apply to any model of any aspect of the physical world" A little humility might be in order here. I would insert YET into that sentence. For example, more than 70 years ago G.H Hardy wrote to the effect that his beloved number theory was pure and unadulterated by military applications (actually any application outside pure math) unlike calculus, say. But his sentiment was premature since number theory underlies modern cryptography which is crucial for military signal processing (and of course e-commerce). Algebraic topology was another "pure math" discipline with little use in the real world. But now persistent homology is an important tool for network analysis and finding patterns in huge data sets. math guy

That's a nice paper. His derivation of Euler's formula is exactly like I used to teach it in calculus. As a small point, and I think he makes this mistake, Euler's formula is e^(ix) = cos x + i sin x, and Euler's Identity is then e^(i•pi) = -1. The first is a formula for any x, and the second is just a fact, like sin 30 = 1/2. hazel

PS: I found an interesting first level discussion: http://travismallett.com/wp-content/uploads/2013/02/MATH-182-Euler_Article.pdf kairosfocus

re 199: Yes. it was my opinion also that "Euler’s Identity itself does not model any physical phenomenon." I don't know whether you've read all the intervening posts, but I've discussed how teaching complex numbers was one of my favorite subjects, and kf and I have both remarked on how useful they are in described some important and complex phenomena. You ask, "Given the nature of the universe, were humans bound to create a math system where EI is true? ... Our best math that contains (as far as we can tell) the essential terms pi, e, i, somehow managed to have such a nature that EI is true. Moreover, the math implied EI before Euler discovered it." First, I agree and have discussed above that one of the features of math is that the things we discover are true whether we have discovered them or not. I'm not sure we can say that humans were "bound to" discover EI, because the history of math, and human civilization in general, might not have gotten there. (For instance, there may be wonderful mathematical topics, tools, and results that we have not developed and discovered yet, and might not ever.) But given making the basic decisions about terms and the work we have done, the EI is a straightforward result of starting with the number system and developing trig, exponentials, and complex numbers. At 70, I wrote,

A common distinction is that mankind has invented the particular symbol systems that we use, but within those systems, once established, the logical consequences are then discovered as inevitable logical consequences. For instance, once the number system was extended to include complex numbers, Euler’s Identity e^(i*pi)= -1 was discovered.

and at 170

As I used to tell my seniors, they walked in to first grade learning the number fact 1 + 1 = 2 and walked out knowing the number fact e^(i*pi) = -1. These facts, and everything in between, given suitable definitions, are logical consequences, contained within the system, that go back to the original structure of the natural numbers.

So as far as all this goes, I think we are in agreement. You write,

I assume that the universe pre-existed humans, and I assume that it has actual properties that can be perceived, so to the degree that our maths describe the real world, the maths have been discovered. This seems obvious. What am I missing?

I think there are two different kinds of things that we can say have been discovered. As a purely mathematical fact, we discovered that e^(i*pi) = -1 as a logical conclusion based on the foundational number system and suitable definitions we developed along the way. This is a logical discovery. However, the fact that complex numbers can be used to model the quantum wave function, for instance, is an empirical discovery: people had to propose a model using complex numbers and test it against empirical evidence to see if it worked. And lots of math has been logically discovered that doesn't apply to any model of any aspect of the physical world, so there has been no corresponding discovery of a workable description of some part of the world that uses that math. hazel

M62, It is obvious that the five numbers in the Euler expression are ubiquitous in many contexts of pure and applied mathematics, the latter engaging the real world in many practical ways. What is perhaps most interesting is that they are exceedingly diverse: 1 and 0 are foundational to the natural numbers, which are indeed core to Math. The circle's geometry defines pi, which is another ancient branch of Mathematics. Then, e and i are at the heart of modern developments and applications since the 1600's, again a very divergent situation. However, the complex plane and complex exponential forms of complex numbers [which are tied to sinusoids and to power series representations) do point to a unit length vector sweeping the unit circle in the complex plane. In that context we may ponder z = 1* [e^i*w*t] and the case where wt = pi rads, this notoriously being "the natural unit of angle." (Yes, even units of angle may not be arbitrary!) Once we see this and re-arrange, we get the astonishing unification: 0 = 1 + e^i*pi. This means that the five key numbers and key operations: sum, product, exponentiation are locked together to literally infinite precision, and that this will apply in any world where at least a 2-d conceptual space is possible (which requires the continuum, thus N --> Z --> Q --> R as sets, then going vector through i*R an orthogonal transform of R, thus C). One take-home is that here we have reason to be confident in the coherence of a wide sweep of Mathematics, something not to be taken for granted post Godel. And this includes the power of transforms to bring differential equations to the complex plane, then by extension difference equations -- instantly applicable to a lot of abstract and applied system dynamics. There is much more, those who would trivialise the result would be well advised to reconsider. Where, of course, all of this is pointing to the way the substance of logical principles of structure and quantity are embedded in possible worlds . . . one of the themes it seems some struggle to accept as material and significant. KF kairosfocus

Hazel: Euler’s identity is neat, and is one of many marvelous math facts, but I don’t think Euler’s identity is embedded in the universe anyplace, nor models any specific phenomena. I might be wrong, though, and would welcome being corrected by an example. Euler's Identity itself does not model any physical phenomenon. But the terms are broadly employed in physical models, and integral to our best and wildly accurate models, such as quantum physics and General Relativity. Our known perceptions and discoveries of the known physical world certainly seem to indicate that pi, e, and i are fundamental in describing the universe's properties. This is not controversial. Maybe our accuracy is off a bit, but we're in the ballpark, no? What is interesting about EI is that these ubiquitously applied, physically accurate terms have such a "simple", elegant, fundamental relationship. Given the nature of the universe, were humans bound to create a math system where EI is true? If not, what a lucky accident. Our best math that contains (as far as we can tell) the essential terms pi, e, i, somehow managed to have such a nature that EI is true. Moreover, the math implied EI before Euler discovered it. Any mathematical or symbolic system has implications that are true whether anyone has perceived them or not. In that sense, anything not fundamental is discovered and not invented. I assume that the universe pre-existed humans, and I assume that it has actual properties that can be perceived, so to the degree that our maths describe the real world, the maths have been discovered. This seems obvious. What am I missing? mike1962

To JAD and others: John Conways game of life and wikipedia's article Lizzie/ febble/ queen penguin used to love that one, too, hazel. I am still not sure how it is relevant to the claim that mathematics permeates the universe because mathematics was used to intelligently design it. Does it really bother people that the intelligent humans merely discovered, rather than invented, these amazing gifts they have given us? To me their ability to tap into the universal information and make sense of it is inspirational. ET

H, I will continue to point out what is plainly well warranted and highly material, despite the all too common tendency to sideline or ignore or even sometimes to try to make such seem false or dubious or to exert inappropriate selective hyperskepticism to dismiss. I will also continue to show that such reflects the balance on the merits. KF kairosfocus

kf writes, "H, I repeat" Yes, you do. :-) hazel

H, I repeat, there is a logic of structure and quantity that pervades our observed cosmos and in part such will pervade any possible world. Our study of the logic of structure and quantity and abstract model worlds we may create in that study in the end will have to be constrained by that substance. For instance, many times now I have shown that distinct identity -- a necessity for any particular world -- directly has the natural numbers as a corollary. We use fingers, sticks, hash-marks etc in simple Mathematics precisely because of the principle that two sets with the same cardinality can be put in one to one correspondence. In short, our discovery of numbers is tied to the embedded reality of same in any possible world. Where too, I have pointed out that more sophisticated abstract logic model worlds are possible worlds, so entities we discover in exploring them which are necessary entities will also be present in all possible worlds. Similarly, if the logic model is sufficiently similar in material respects, some things in that model will be in the world we inhabit. The utility of Mathematics traces to these factors. KF kairosfocus

JAD, I have emphasised the contrast between the substance of the logic of structure and quantity and our study of it. KF kairosfocus

EG, your comment was on a series of OP's I have been developing on logic and first principles. KF kairosfocus

JAD writes, "So what the issue now with games? I agree games are something we invent, and mathematicians have long been fascinated with them. So?" The Game of Life is not really a game. It's an iterative model that incorporates, at a simpler level, the same principles that produces fractals such as the Mandelbrot set. It's a mathematical system. If you're not familiar with it, I suggest that you find out some about it. hazel

kf, you write, "Much of modern mathematics drew its initial impetus from considerations of quantitative and structural phenomena we encountered in the world. Axiomatisation is subsequent to that and is materially conditioned by it" Yes, I agree, and just wrote Mike about that point. You write, "the two cannot be severed neatly by saying things like, one is studying “pure” Mathematics without reference to particular applications." I don't know why not. There is a great deal of math that has no particular application, and a great deal of math that was developed for the pure satisfaction of drawing out logical conclusions. Of course, a great deal of math can be applied to real world situations, and we often have found that math that has been discovered in the context of pure math is a good tool for describing some aspect of the real world. I have always agreed with those who point out that the ability of math to accurately describe and model the world is a fact. However, as the Einstein quote I offered earlier said, our descriptions and models are always only abstract approximations of physical reality, and the certainty that we have about pure mathematics does not carry over to our mathematical models of reality. hazel

Hazel @ 170 wrote: “The [game] of Life is clearly an invention. I can’t believe that these particular rules, as well as all other possible rules for similar games, eternally exist separate from anyone ever knowing about them, like a circle.” Very early on in the discussion which started on an earlier thread. I wrote the following:

Everyone involved in this discussion are missing some obvious points. As Kf pointed out, my point up @ #40 is that numbers have properties and those properties have been discovered not invented. So how can anyone claim that math (or if you’re a brit, “maths”) is a human invention? That’s not to say that human’s didn’t invent the symbolism used to do mathematics. But without the existence of mathematical truth there would be no reason for the symbolism. Furthermore, the development of mathematics preceded the advances of the physical sciences by centuries.

https://uncommondesc.wpengine.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/#comment-669650 I could take it a step further and say that we that we invent number systems like Roman numerals or Greek numerals and now Hindu-Arabic numerals which use different symbols. You can easily invent your own number system. Here’s mine: ! @ # $ % ^ & * ( ). I just substituted those symbols for the numbers 1-10 by pressing the shift key on my computer’s keyboard. We could actually do math with those symbols. (Thanks but thanks.) Does anyone on my side dispute that we can and do invent number systems? However, could we invent a number system that did not have primes or irrational numbers? If you think we could, go ahead and invent one. The next question would be, would it serve any practical purpose in the real world? So what the issue now with games? I agree games are something we invent, and mathematicians have long been fascinated with them. So? It doesn’t follow logically that if there some things we invent and use as mathematical tools that that is all math is-- something we invented. We have invented radio telescopes to explore the universe but it doesn’t follow that everything we have discovered so far are just artifacts of the radio telescopes electronics. We invented astronomical tools because we could see something out there that we wanted to explore and understand. john_a_designer

KF

My original discussion which sparked this Op highlights that.

I thought that it was one of my comments that sparked this OP. After all, my name is in the title. :) Ed George

Those are good questions, Mike. Let me answer the second question, and then say a bit more about the first. Our pure math has been first motivated, historically, by symbolic representations of simple things in the world around us. kf and I discussed how pebbles of fingers motivated the concept of the unit 1, and it's easy for me to see how simple things like a rope pulled tight, or a rope swung around a stake, motivated the concepts of a line and a circle. I would expect similar experiences would have motivated the basis of the alien's math. However, even assuming they were sophisticated enough to get here, they probably wouldn't have the exact same math, or know the same things we do. It's very likely they wouldn't have the Game of Life, and they might not know about fractals or the Mandelbrot set, for instance, or perhaps all the things we've learned about primes. On the other hand, they may have developed some techniques we don't have: maybe they've proved Fermat's Last theorem or ways of directly evaluating some integrals that we can only approximate. The point is that once the basics are set, it's a contingent matter as to what paths of discovery they might have taken. This is all very hypothetical, so it's easy to make stuff up, but it's interesting to think about. hazel

H, I would suggest that in its essence Mathematics -- pure and applied alike -- is about the logic of structure and quantity. The core substance of that is embedded in the world as in effect rational principles of reality that are partly intelligible. In our study of the logic of structure and quantity -- the discipline of Mathematics -- that embedded structure tied to requisites of distinct identity such that there is a world, powerfully influences our work. So, the two cannot be severed neatly by saying things like, one is studying "pure" Mathematics without reference to particular applications. Besides, much of modern mathematics drew its initial impetus from considerations of quantitative and structural phenomena we encountered in the world. Axiomatisation is subsequent to that and is materially conditioned by it. KF kairosfocus

hazel: But, in the long run, our math would be the same. Why? mike1962

The same, basically. More advanced math might have different approaches, because we know that the same results can sometimes be expressed in different systems. But, in the long run, our math would be the same. hazel

hazel, if some extra-terrestials showed up here, would you expect that their fundamental mathematics, such as pi, e, i, and the relationship thereof, would be the same as ours or different? mike1962

wjm asks a good question, "I have another question for you, which you may have already answered and I missed it. Can you “not accept” a platonic realm at all, or is it just the idea of a platonic realm that contains an infinite amount of information (including future information) that you cannot accept?" It's easy to accept that the definition of a circle invokes, somehow (it's the somehow that I don't know) a perfect circle, that it's properties therefore exist as immediately existing in relationship to that original definition, and that we discover those properties through the process of proof through deductive reasoning. I used to love to teach this idea in geometry: it's a critical part of intellectual development to learn about the power of deductive reasoning. However, I can't really grasp a sense of there being some Platonic world in which the circle exists, because, as I am trying to explain, once you do that you open up the door to all and every mathematical system, including ones I could invent today, also existing Platonically. So despite how obvious Platonic reality seems in respect to the circle, it doesn't seem to work for me if carried to its conclusion. Obviously, I don't know what works to explain why logical conclusions appear there for our discovery, true whether we ever discover them or not. Everything existing in a Platonic realm isn't a satisfactory explanation for me. hazel

kf , my post was about pure math. I didn't address the issue of how math can be applied to describing the real world. hazel

hazel, As I said to Ed George, whether or not anyone here finds any particular argument "compelling" is irrelevant in terms of offering arguments and rationally criticizing them. I have another question for you, which you may have already answered and I missed it. Can you "not accept" a platonic realm at all, or is it just the idea of a platonic realm that contains an infinite amount of information (including future information) that you cannot accept? BTW, everyone: Keep this discussion REAL friendly and REAL respectful, please. This is about the logic, evidence and arguments, not about convincing hazel or anyone else of anything or about "revealing" anything about anyone who disagrees. William J Murray

hazel:

This is a perennial philosophical issue that will probably never be settled.

We disagree. How can you tell if the Game of Life was invented vs discovered? ET

H, Does Mathematics address an objective substance of structure and quantity? My original discussion which sparked this Op highlights that. As a good example, can we justly imagine that we could go back and invent a new game where 2 + 2 = 5 say, and be just as successful in the context of utility and reliability? I also beg to remind you that God is a philosophical issue, not "merely" a matter of religious dogma. In context, there is in fact a longstanding view that the kernel of truth in Plato lies in the mind of God, a point Pruss actually brings up in his thesis on possible worlds, which is linked.KF PS: The overton window may not be familiar to you, but it is most definitely not bizarre; it is what is driving the wider context of the ongoing civilisational civil war. Absent that conflict, the issue of a substantial core to Mathematics (of all things) would not be up for debate. PPS: I pointed out that the historical invention of artificial solutions to quadratics were not the only or the most fruitful way to approach complex numbers. This responds to a point you raised about said numbers. kairosfocus

Thanks for all the responses. Here are some replies in approximate order of relevance wjm's comment at 174 is most relevant:

Hazel expressed part of the dilemma this way: "So I don’t know what to think here. I can’t accept the idea that all possible consequences of all mathematical systems eternally exist in some Platonic realm, even ones like the Game of Life that has just been invented.” Referring to the part in bold, why is that, Hazel? Why is it that you “can’t” accept it?

The Game of Life illustrates the problem. This was just invented 50 years ago, and there is nothing to make me think it had to be invented. And yet, once it was invented, a infinite amount of new logical consequences were all of a sudden waiting to be discovered. In addition, those consequences can only be discovered by actually stepping through the generations: there are no algorithms that can predict a result. Furthermore, I'm pretty sure I could invent a game somewhat like Life today (I would probably make it on a hexagonal rather than rectangular grid), and in doing so make a whole new infinite set of logical conclusions. It makes no sense that all these results could have been pre-existing eternally in some non-material Platonic world. Perhaps you can accept that as the case, or possibly the case. I can't. Given that we are speculating about something that can't possibly be experienced, I think my non-acceptance and your acceptance (if in fact this is what you believe) are on the same footing. As I have repeatedly said, I can see validity in various views, but nothing that seems compellingly conclusive. The goal of my post was to point out some of the aspects that most make it mystery to me, and to discuss some of the aspects of the discover/invention issue. ============= At 173, kf writes, "the “discovery” issue is in material aspects driven by the substance of structure and quantity embedded in the world", and then goes on to talks about vectors in the real world. (He also again talks the real world at 175.) However, I made it clear at the start of my post that I was just talking about pure math, so the rest of his comments aren't relevant. ============= 175 kf writes, "WJM, no-one is suggesting that “all possible consequences of all mathematical systems eternally exist in some Platonic realm.” wjm asks me why I couldn't accept that possibility, so at least implicitly he was suggesting it as a possibility. kf writes, "That said, there is a credible candidate who can hold “all possible consequences of all mathematical systems eternally exist[ing] in some Platonic realm [a certain Mind].” God." Hmmm. Earlier, kf seemed to chastise me for referring to religious explanations, and now he says that an explanation here is that the Mind of God could be the place where all the infinite number of possible consequences of Conway's Game of Life have eternally resided. My comments to wjm about a Platonic realm apply here. Invoking the Mind of God as a solution doesn't solve the mysteries for me. ============= At 173, kf also writes, "[I need not elaborate on the damaging results of playing overton window lockout games driven by all too common behaviour patterns taught by Alinsky and other cultural marxists.)" I have no idea what this bizarre comment is about, or what precipitated it. How this discussion could possible have anything to do with Marxists is a mystery. ============= to JAD at 177: You write, "So Hazel’s personal incredulity and belief is what settles the argument?" Absolutely not. I've never claimed that I'm trying to make a logical argument for what is the case. I've clearly said I can see validity in various points of views, as well as flaws, and that I am basically baffled about the situation. I don't think anyone can make a strongly compelling deductive argument about what is in fact the case without invoking premises that are themselves open to the very questions we are interested in. This is a perennial philosophical issue that will probably never be settled. hazel

Hazel @ 170: “I don’t know what to think here. I can’t accept the idea that all possible consequences of all mathematical systems eternally exist in some Platonic realm.” [Characterizations about the character or mind of others will be edited out. - WJM] Deductive arguments must be based on premises that are either self-evidently true or probably true. If it’s the latter, premises which have a higher probability of being true generally make for a stronger argument. john_a_designer

hazel:

If we discover math, then that implies what we discover already exists. But how? In what sense does all of math exist, waiting to be discovered.

It's part of the intelligent design of the universe. All of the information required by the universe is here for us to tap into. Giuseppe Sermonti touched on this in the chapter "I can only tell you what you already know" in the book "Why is a fly not a horse?" ET

WJM, no-one is suggesting that "all possible consequences of all mathematical systems eternally exist in some Platonic realm." It is shown that for a distinct world W to exist it must have distinct identity based on defining characteristics. Thus W = {A|~A}. The emptiness of the border and the contrasted set W' (excluded middle), the unity of A and the duality of A and its complement show us that once W is, 0, 1, 2 necessarily obtain. But also, we see the von Neumann succession of order types so, this extends to the naturals, necessarily. We thus see that certain core structures and quantities obtain in any world, they are necessary entities. Others will obtain in particular worlds but not others, contingent entities. That said, there is a credible candidate who can hold "all possible consequences of all mathematical systems eternally exist[ing] in some Platonic realm [a certain Mind]." God. KF PS: This suggests that H's declared lack of interest in the logic of being thus necessary vs contingent entities and possible worlds may well be part of the problem. kairosfocus

I'm impressed by hazel's clear thought leading up to the dilemma she expressed in #170, and her ability to actually see that dilemma clearly. Hazel expressed part of the dilemma this way: "So I don’t know what to think here. I can’t accept the idea that all possible consequences of all mathematical systems eternally exist in some Platonic realm, even ones like the Game of Life that has just been invented." Referring to the part in bold, why is that, Hazel? Why is it that you "can't" accept it? William J Murray

H, I think your last comment is significant, though the "discovery" issue is in material aspects driven by the substance of structure and quantity embedded in the world. The contrast, study vs substance is really important. I suggest that even when there was invention, there may well be discovery that was always there. In the case of complex numbers [not just "imaginary"] you wrote:

there can by a hazy line between invention and discovery. At some point, someone (the history is more complicated, really) decided to consider the square root of a negative as a number, not an impossibility, and named the sqrt(-1) as i. This was an invention – a definition, in some sense. But one could argue that it was an inevitable logical step, and someone just had to decide to follow it, in which case we might consider the concept that the sqrt(negative) is a legitimate number is a discovery, and only the symbol was an invention. Retrospectively, this turned out to be a great idea, leading to complex numbers, which we have seen are of great importance.

I think the history is misleading us. We live in a spatial world where vectors are natural and they can rotate. For basic example, conservation of angular momentum and interaction with a gravity field produce precessional effects. There is a 23,000 y precession of our planet's orbit. In that context, it is significant to note that complex numbers are vectors, and that a very natural path to seeing the legitimacy of sqrt(-1) is the i* right angle A/C rot operator approach. And yes I am using operators as a mathematical structure. Consider:

(1,0) along ox --> call it x1. i*x1 --> (0,1) along oy. do i* again: i*i*x1 --> (-1,0) along -ox. in effect, considering ox as reals line, i*i*1 --> -1 Thus, reasonably i^2 = -1 So it makes sense to identify i with sqrt(-1)

Is rotation of a vector of length in a specific direction "natural"? Simply put a hole near the end of a 1m rule, put it on an axis and rotate it. Rotation is a property of a world with space. Angle is also a naturally manifest quantitative structure. So, picking a particular rotation of interest and exploring it is experimenting with and exploring properties, not arbitrary, free invention. It is constrained by relevant structures and quantities of space, physical or conceptual. (Recall, necessary entities or properties may be identified in an abstract model world but as such is a possible world the necessary entities will obtain for all worlds including this one. That already highlights how important logic of being can be.) Of course, if I take your declarations above literally, you may well not read this. EG, I am pretty sure, will not [I need not elaborate on the damaging results of playing overton window lockout games driven by all too common behaviour patterns taught by Alinsky and other cultural marxists.) So, I note for record, on the merits of the case. KF kairosfocus

EG:

if we stick with Pi and prime numbers, which we all agree have some peculiar and interesting properties. In both cases, they only exist because we have established conditions for them to exist.

The relevant structures, quantities and properties existed in the world before we thought of them. That recognition of fact is pivotal and it is revealing that you still struggle to distinguish between the substance and the study of structure and quantity. KF kairosfocus

Hazel@170, again a much clearer presentation of the issues than anyone else here has been able to do. But if we stick with Pi and prime numbers, which we all agree have some peculiar and interesting properties. In both cases, they only exist because we have established conditions for them to exist. A number only divisible by itself and one. And the ratio of circumference and diameter of a perfect circle, a perfect circle that only exists in our imagination. So, the question then becomes, is any mathematics derived from these two invented concepts inherent in the universe, or simply the consequence of our inventions? I think we all agree that the concept of unity is the foundation of the universe, and of all mathematics. One exists. My expertise is chemistry which, I think we can also agree, involves complex relationships that can be modelled using mathematics. But, fundamentally, all we are dealing with is hydrogen with the addition of various numbers of protons, neutrons and electrons that combine with different types of bonds. I know, it is far more complicated than this but most of our chemical reactions can be modelled using this simple framework. Does that mean that the complex mathematics needed to model these reactions are inherent to the universe? Or are they inventions that we derived from that fundamental aspect of math (one) and manipulated to represent what we observe (or, model)? Or, have I once again stepped way beyond my comfort zone and just blowing smoke out my anal orafice? :) Ed George

I want to offer some comments on the discovered/invented issue. First a few disclaimers/reminders. This is a perennial issue, and I can see the situation from various viewpoints. I want to make some observations and ask some questions without having definite answers on some issue. Also, I want to talk just about pure mathematics. By pure math I mean math that exists within a symbol system, in which conclusions are logically developed within that system, and built from previously established aspects of the system. For instance, I believe we have agreed that the body of math we learn up through high school starts with natural numbers, which start with positing a unit number 1, defining successors, and building from there. As I used to tell my seniors, they walked in to first grade learning the number fact 1 + 1 = 2 and walked out knowing the number fact e^(i*pi) = -1. These facts, and everything in between, given suitable definitions, are logical consequences, contained within the sytem, that go back to the original structure of the natural numbers. Now what can we say about what was invented and what was discovered here? I think is uncontroversial to say that specific symbols, notation systems, and terms are "invented", although "created" might be a better word, or "developed" for things which time and the input of multiple people to become adopted. Two examples mentioned in previous posts was the invention of the Arabic number system, with zero as a placeholder, and the notation system for the derivative. On the other hand the bulk of mathematics has been discovered as a logical consequence of aspects that have been previously establish. kf offered an interesting example. Given the definitions of even, odd, prime, factor, and multiple, we can prove that for any prime p > 3, p^2 - 1 is a multiple of 24. That is not obvious, and it is a discovery, not an invention. It is easy to think that all the logical consequences that build up in our math system are, in some sense, already there when we discover them, and were just as true before we even thought of them, much less proved them, as they are after we have proved them. With all this said, here are some examples that bring up points that puzzle me a bit. First, there can by a hazy line between invention and discovery. At some point, someone (the history is more complicated, really) decided to consider the square root of a negative as a number, not an impossibility, and named the sqrt(-1) as i. This was an invention - a definition, in some sense. But one could argue that it was an inevitable logical step, and someone just had to decide to follow it, in which case we might consider the concept that the sqrt(negative) is a legitimate number is a discovery, and only the symbol was an invention. Retrospectively, this turned out to be a great idea, leading to complex numbers, which we have seen are of great importance. Earlier I said, "It is easy to think that all the logical consequences that build up in our math system are, in some sense, already there when we discover them" But in what sense is the part that puzzles me. It is common to consider some concepts, such as that of a perfect circle, as eternally existing as an ideal Platonic form, irrespective of whether there is anyone anyplace to know of it. But, as mentioned earlier in this thread, what about the 1 millionth digit of pi? What about the billonth? What about expansions of pi to different bases, such as the 1 billionth digit to base 2000? All of the numbers are certainly what they are irrespective of whether anyone will ever know them. Are they all part of some Platonic realm? I can't believe that such numbers only exist when they are actually figured out by someone, but also find it hard to believe that every conceivable mathematical consequence (all these digits of pi, or the state of every single point under the Mandelbrot set rules) already exists. I do not have a clear picture of what the case could be. Here is an example that is even more baffling. Consider Conway's Game of Life. (See https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life if you're not familiar with it.) The ggme of Life is clearly an invention. I can't believe that these particular rules, as well as all other possible rules for similar games, eternally exist separate from anyone ever knowing about them, like a circle. And yet, once you set a beginning configuration in generation 0 (gen0), the resulting history of the space is completely determined. Furthermore, the only way you can discover the state of any particular generation, such as that at gen100, is to actually let the process run for 100 generations. You can't "figure it out" without just letting it happen. In what sense do all histories of the infinite number of possible beginning configurations exist before they are actually instantiated? So I don't know what to think here. I can't accept the idea that all possible consequences of all mathematical systems eternally exist in some Platonic realm, even ones like the Game of Life that has just been invented. On the other hand, I can't believe that they exist only when someone instantiates them in writing, as clearly they are what they are - the outcome is logically inevitable whenever we decide to pursue a particular issue (digit of pi, generation of Life, etc.) If we discover math, then that implies what we discover already exists. But how? In what sense does all of math exist, waiting to be discovered. I'm baffled! :-) hazel

Cross-threading from LFP4, no. 101 as also highly relevant: Pruss’ thesis: http://alexanderpruss.com/papers/PhilThesis.html — on possible worlds. KF kairosfocus

F/N: Let me pull forward JAD's note on and from Penrose:

[JAD, 12 above:] According to mathematician Roger Penrose, who collaborated with Stephen Hawking on some of his early work, “mathematics seems to have its own kind of existence.” He then goes on to explain:

It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures… It also has this relationship to mentality because we certainly have access to mathematical truths. I think it is useful to think of the world as not being a creation of our minds because if we do then how could it have been there before we were around? If the world is obeying mathematical laws with extraordinary precision since the beginning of time, well, there were no human beings and no conscious beings of any kind around then. So how can mathematics have been the creation of minds and still been there controlling the universe? I think it is very valuable to think of this Platonic mathematical world as having its own existence. So let’s allow that and say that there are three different kinds of existence. There may be others, but three kinds of existence: the normal, physical existence; the mental existence (which seems to have, in some sense, an even greater reality – it is what we are directly aware of or directly perceive); and the mathematical world which seems to be out there in some sense conjuring itself into existence – it has to be there in some sense. https://www.reasonablefaith.org/media/reasonable-faith-podcast/roger-penrose-interview-part-1/#_ftn3

Just to clarify, earlier in the interview Penrose described his metaphysical world view as a tripartite one consisting of the physical world, the mental world and a separate and distinct mathematical world. He goes on to explain that… ’there is the relationship between these three worlds which I regard, all three of them, as somewhat mysterious or very mysterious. I sometimes refer to this as “three worlds and three mysteries.” Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision. That’s mystery number one.’

While contexts of approach differ, convergence is plain. For simple example, eternal character will be a characteristic of successful candidate necessary beings. Part of the context of what begins/ceases has a cause. KF kairosfocus

Found it: it was in a comment by john_a_designer, #12. hazel

hazel @164: No I did not. William J Murray

H, again, I am left chuckling. One can see why by scrolling up. Never mind, the substantial matter is settled on the merits: there is every reason to acknowledge that this world and indeed any possible world will embed a substantial core of quantity and structure starting with the naturals. A spatial world will have instantiated continuum. Bring in time and rates and accumulations of change become material. Look at the notoriously mathematical ordering of the physical world we observe and note the fine tuning which supports C-Chemistry, aqueous medium cell based life. All of this then does raise meta issues, and a highly intelligent, exceedingly rational and mathematically competent designer of the cosmos becomes a serious matter. But pulling back, the way structure and quantity pervade the world is clearly a profound issue, one well worth our attention. KF kairosfocus

Also, wjm, while you're here, I'm curious: didn't you have a quote by Penrose in your original OP, or am I confusing it with another post. If so, could you post it here? I found it interesting, and would like to refer to it. hazel

[Questions or comments about how I moderate my own threads will be deleted. - WJM] hazel

@WJM obvious implication is intelligence did it. Physical mechanisms cannot generate information regarding abstract concepts. Intelligence is the only thing that can physically instantiate abstractions. EricMH

I see no one has come forth with an argument that mathematics are invented by humans, and there is no apparent rational criticism against the argument that mathematics are discovered, inherent aspects of the universe. I think the question now becomes: what can we infer from the fact that mathematics is an inherent aspect of the universe and of our existence? William J Murray

MG, yup. But, it seems many insist on going down such roads. About par for the course of our civilisation today. KF PS: Your testimony (which reflects what is often talked about as being right-brained) is interesting; especially as recognition of structural patterns can come long before being able to weave a string of sentences in some logic world's language to explicate the insight:

Quine’s “refutation” of mathematical platonism is a long syllogism of form If A-mat is true, then ….., then….., then platonism cannot be true. But the initial premise is not actually stated, only subconsciously assumed as true (all the “cool kids” know A-mat is true and need not be mentioned). But my own personal experience (flashes of mathematical insight out of the blue that are invariably true, although formal proof only comes weeks or months later) demonstrate that platonism is valid. Thus the contrapositive of the above syllogism falsifies A-mat. Unfortunately I may not be able to persuade everyone using my own subjective experience. However J. Hadamard chronicles many other prominent mathematicians having “aha” glimpses of platonic reality in his small book “The Psychology of Invention in the Mathematical Field”. (In the preface, Hadamard suggests that “discovery” might be a better term than “invention”.)

Sounds like a degenerate research programme is hampering sound progress. In this case the logical upending of the chain of implications is a telling one-liner. kairosfocus

ET, 153: you seem to have spotted a key problem, cf. my just above. KF kairosfocus

H & EG: Chuckle, again. It seems the real problem has surfaced. You are engaging in essentially objecting commentary on a series of articles that are engaging logic and first principles of reason (including how this leads to numbers etc as necessary features of any possible world) but have no interest in either the substance or the argument, while having fairly obvious disdain for the presenter and lack of awareness of key ideas involved. No wonder there is little motivation to follow the substantial interest or concern that there may be key issues of truth on the table. That readily explains why objections and dismissive comments so often turn the substance into a strawman caricature and tend to be distractive and dismissive. In addition, you are unfamiliar with the nature of key issues and concepts but are not engaging with what would help you begin to change that unfamiliarity. This then leads to the projection of hostility or confusion or incoherence etc, especially when this little spermologos has the temerity to attempt to explain or correct or point out fallacies. I am left shaking my head and chuckling. Perhaps, we can start afresh on a different footing? KF PS: In case of some interest, I draw notice to Internet Enc of Phil on PW and logic of being, which would at least help you appreciate that I am speaking to substantial issues that are foundational to reasoning about being, logic, Mathematics, Science etc:

3. The Necessary/Contingent Distinction A necessary proposition is one the truth value of which remains constant across all possible worlds. Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false. The necessary/contingent distinction is closely related to the a priori/a posteriori distinction. It is reasonable to expect, for instance, that if a given claim is necessary, it must be knowable only a priori. Sense experience can tell us only about the actual world and hence about what is the case; it can say nothing about what must or must not be the case. Contingent claims, on the other hand, would seem to be knowable only a posteriori, since it is unclear how pure thought or reason could tell us anything about the actual world as compared to other possible worlds. While closely related, these distinctions are not equivalent. The necessary/contingent distinction is metaphysical: it concerns the modal status of propositions. As such, it is clearly distinct from the a priori/a posteriori distinction, which is epistemological . . .

I have extended this sort of thought to addressing being, as in why would a necessary entity exist in all worlds? Plainly, because it is part of the framework for any world to exist. Distinct identity is such a feature as a distinct possible world would be by contrast with others. This brings up the partition W = {A|~A} which then has as corollaries LNC and LEM, also nullity, unity, duality. Succession of order types then delivers the naturals and instantly the first transfinite. Numerical structure and quantity are embedded in the framework for any possible world. Which, makes sense of the utility of Mathematics in ours. kairosfocus

mike @ 115 Yes my point is that Quine's "refutation" of mathematical platonism is a long syllogism of form If A-mat is true, then ....., then....., then platonism cannot be true. But the initial premise is not actually stated, only subconsciously assumed as true (all the "cool kids" know A-mat is true and need not be mentioned). But my own personal experience (flashes of mathematical insight out of the blue that are invariably true, although formal proof only comes weeks or months later) demonstrate that platonism is valid. Thus the contrapositive of the above syllogism falsifies A-mat. Unfortunately I may not be able to persuade everyone using my own subjective experience. However J. Hadamard chronicles many other prominent mathematicians having "aha" glimpses of platonic reality in his small book "The Psychology of Invention in the Mathematical Field". (In the preface, Hadamard suggests that "discovery" might be a better term than "invention".) math guy

KF @ 137 (sorry about the delay, I have a math-related day job that doesn't allow posting to blogs) I agree that rejecting LEM leads to cacophony, chaos, and confusion. Although Brouwer, Heyting, and the other Intuitionists recover a substantial proportion of classical analysis without LEM, it is not sufficient to model QM (for instance). math guy

Oh, I can see where kf and I are in the sense that he has big philosophical ideas that are important to him, but not to me, and I'm just talking about practical, everyday issues about the nature of math. I think ET is right in this regard. [Unnecessary personal contented deleted - WJM] Now I see he's posted two posts about "possible worlds" on one his other threads. I have no idea if anyone here cares to read those, or not. hazel

Hazel, do you know what ET is talking about talking past each other? :) Ed George

In an eagerness to "one-up" the other person it is very common to talk past each other. Just sayin'... ET

:-) hazel

[Posts that are entirely about personalities and no substance will be deleted - WJM] Ed George

[Posts that are entirely about personalities and no substance will be deleted - WJM] hazel

[Posts that are entirely about personalities and no substance will be deleted - WJM] Ed George

kf, listen to this: I am NOT INTERESTED in your posts. I can't even imagine wading into a discussion with you about them. They don't make much sense to me, and what sense I do make of them are about approaches I am not interested in. Can't you just accept that? [You are not obligated to respond to any post. If you don't wish to interact with someone posting, don't. - WJM] hazel

H, chuckling again. This one is called turnabout projection. Let's notice the nice little bracket: " I tried reading your post that you linked to, and I’m sorry to say it’s way too much of a hodge-podge of concepts (including some religious ones) and disparate images and notations for me to get any coherent picture . . . " (Notice the close association between religion and incoherence?) I took time to note that you confused a philosophical usage for a religious one -- where it is notorious that that word is loaded language in this sort of context. Notice your onward "I see God – that says religion to me" -- red flagging confirmed. Next, I sufficiently showed the logical structure that your dismissive assertion "hodgepodge" falls to the ground. My use of a few symbols is well within reasonable use. I think the PW concept is sufficiently outlined, is clearly useful and relevant. As for projecting accusations, several times you have used dismissive rhetoric including red herrings -- ponder "religion" as an irrelevant red flag -- and strawman fallacies -- ponder the hodgepodge picture you painted and the actual coherence I showed. It is entirely appropriate for me to point such out and ask for a more serious tone. KF kairosfocus

kf, you started off by writing

First, the attempt just above to red flag mistakes mention of the God of Ethical Theism — a philosophical concept — for a religious one. Obviously, with all the invited disdain in a world full of aggressive secularists — it is the crowning piece.

[Posts that are entirely about personalities and no substance will be deleted - WJM] Then you write,

(Pardon, but I think on fair comment it is already well within my rights to point out to you that you should reconsider dismissive rhetoric which has kept on cropping up.)

As far as I can tell, my "dismissive rhetoric" is that I can't follow all your arguments, and I'm not very interested in whatever level of philosophy you are interested in. I tried to read the post because you asked me to. Your interests and style are fairly esoteric, and a bit eccentric, and they don't appeal to me. I made an effort, and told you I couldn't get into it. That's all. And then you write,

(Yet again I find it appropriate to speak on fair comment: you inappropriately tried to set up a strawman caricature, push it over and avoid addressing a substantial issue. Please, stop.)

Here I have no idea what you're talking about. Again, what I did was told you that I wasn't interested in the approach you are interested in, and found your post hard to follow. I don't know what "strawman caricature" you are talking about. I told you I am interested in understanding this world, not philosophical ideas about all possible worlds, so all that you have written about doesn't interest me. Can't you accept that without accusing me of a whole bunch of stuff? hazel

H, I chuckle again. First, the attempt just above to red flag mistakes mention of the God of Ethical Theism -- a philosophical concept -- for a religious one. Obviously, with all the invited disdain in a world full of aggressive secularists -- it is the crowning piece of your dismissal of an alleged hodge-podge. What is noted in passing long after a discussion of possible worlds that gives more details than above, is that God is a serious candidate necessary being and that such candidates will be impossible of being or else actual. Next, the focus is causality and linked logic of being, which brings in the possible world concept (and the nearby world one). Observe the bridge:

some proposed things such as a square circle are not possible of being: there is no possible world in which they would exist as core characteristics [squarishness vs. circularity] stand in mutual contradiction. This already shows LOI and LNC in action, the principle of distinct identity truly is of central importance. [--> BTW, that is a significant result already] Other things are possible of being (some of which are actual). Where, we may contrast that some must exist as part of the framework for any world to exist (e.g. numbers) — necessary beings. Other things may exist in certain possible worlds but would not exist in others, hence: contingent. [--> notice, necessary and contingent beings are identified and set in a PW context] Contingent, on what? Causes. [--> the focal issue for that OP] But first, just what is a possible world? [--> as in, we need a preliminary detailing]

No "mixture of dissimilar ingredients; a jumble," just a logical exploration of important but likely somewhat unfamiliar ideas. Notice, how Wiki is used to provide a handy lowest common denominator definition, duly highlighted in red: "For each distinct way the world could have been, there is said to be a distinct possible world; the actual world is the one we in fact live in . . ." Should sound familiar. This leads to: " the modal status of a proposition is understood in terms of the worlds in which it is true and worlds in which it is false." Modal logic, being a fairly significant context of contemporary logical discussion. I then speak in my own voice:

We may summarise, a possible world is a description of the way the — or, a — world might be, inter alia requiring coherence and sufficient completeness for purposes of analysis or action. For example, in mathematics we routinely construct axiomatic systems that lay out complex abstract model worlds even though, post-Godel we know that no sufficiently complex scheme can be both utterly complete and coherent. Also, that for such schemes we cannot construct an axiomatic system that is demonstrably coherent. (That is, in the end, our confidence in the coherence of our mathematical systems is supported rather than demonstrated; i.e. inductive reasoning is inescapably involved in the practice of mathematics.)

That definition is simply not a confused jumble of incongruities. It leads on to application to how mathematical systems set up model worlds -- i.e. possible worlds -- but if sufficiently complex Godel's strictures apply. (Pardon, but I think on fair comment it is already well within my rights to point out to you that you should reconsider dismissive rhetoric which has kept on cropping up.) Let's go on.

yes, the utility of Mathematics and its application through computing systems is never far from the surface in our ongoing considerations. Where yes, that means that to some extent we must accept the sufficient reality of a host of abstract, logic-model worlds that we may apply them in our thought and even practical work.

I think that is well within my epistemic rights. Now, I returned to the focus on causality. In so doing, let me now bring up a further excerpt from Wikipedia, which gives some context:

from the 1960s onwards – including, most famously, the analysis of counterfactual conditionals in terms of "nearby possible worlds" developed by David Lewis and Robert Stalnaker. On this analysis, when we discuss what would happen if some set of conditions were the case, the truth of our claims is determined by what is true at the nearest possible world (or the set of nearest possible worlds) where the conditions obtain. (A possible world W1 is said to be near to another possible world W2 in respect of R to the degree that the same things happen in W1 and W2 in respect of R; the more different something happens in two possible worlds in a certain respect, the "further" they are from one another in that respect.) Consider this conditional sentence: "If George W. Bush hadn't become president of the U.S. in 2001, Al Gore would have." The sentence would be taken to express a claim that could be reformulated as follows: "In all nearest worlds to our actual world (nearest in relevant respects) where George W. Bush didn't become president of the U.S. in 2001, Al Gore became president of the U.S. then instead." And on this interpretation of the sentence, if there is or are some nearest worlds to the actual world (nearest in relevant respects) where George W. Bush didn't become president but Al Gore didn't either, then the claim expressed by this counterfactual would be false. Today, possible worlds play a central role in many debates in philosophy . . .

In my own voice:

We may thus proceed to understand causes and causal factors, first in a fairly narrow sense: where a contingent entity A would exist in world W1 but would “just” not exist in a closely neighbouring world W2, the difference in circumstances C(W1 – W2) = f1 allows us to confidently identify f1 as among the relevant causal factors that enable A to be. Then, we may explore across several neighbouring worlds W2 to Wn, identifying a broader cluster of factors {f1, f2, . . . fn} such that they are each necessary for and are jointly sufficient for A to be. As an example, ponder the extended fire triangle, the fire tetrahedron . . . . We are also seeing here the significance of experimental studies, observational studies, use-cases, Monte Carlo modelling and statistical investigations, where in effect we set up micro-worlds and study properties as we vary circumstances or ponder natural variations. In this context, we are already clarifying cause.

I then turned to Wiki, again as lowest common denominator:

Causality (also referred to as causation,[1] or cause and effect) is what connects one process (the cause) with another process or state (the effect), where the first is partly responsible for the second, and the second is partly dependent on the first. In general, a process has many causes,[2] which are said to be causal factors for it, and all lie in its past (more precise: none lie in its future) . . .

In my own voice:

Obviously, a specific contingent circumstance — e.g. the unfortunate burning down of a specific classmate’s house on a particular day in 1976 — had particular distinct causal factors summing up to its specific cause (of interest to the Insurance company) etc. [--> I could name the classmate] However, once we loosen to a house burning down [--> I am pointing to a more generic sense, in effect the sort of patterns that at grand level lead to causal laws of nature], we see that we can properly take cause in a looser sense (e.g. of interest to those writing fire safety regulations). This them makes good sense of sufficient but not necessary causal factors as for instance Mackie raised. Obviously, for an event E, all necessary causal factors (in this looser sense) must be present, as knocking out any one will block it. [--> see the fire tetrahedron and how firemen battle house fires] Likewise, a sufficient cluster must be present which may include broader contributory factors. [--> bringing in Mackie] For example, while a court building here could have caught fire through a short, fire fighters and the police were very interested to observe evidence of accelerants. Not all fires are arson, but some are [--> A specific case here that was in the news] . . .

(Yet again I find it appropriate to speak on fair comment: you inappropriately tried to set up a strawman caricature, push it over and avoid addressing a substantial issue. Please, stop.) It is obvious that there is a coherent thread in the relevant OP, and that it sets PW analysis in context, indicating its importance. Let us now take PW analysis as reasonably and responsibly in play and proceed. KF kairosfocus

Hi kf. I tried reading your post that you linked to, and I'm sorry to say it's way too much of a hodge-podge of concepts (including some religious ones) and disparate images and notations for me to get any coherent picture of what it's trying to say, except perhaps that numbers have to be what they are. I think you've lost me on whatever your series of posts is about. At least I tried. hazel

H, a possible world is a sufficiently detailed description of how this or another world might be (for purposes under consideration); usually considered as a set of propositions. In effect were the world defining propositions accurate to reality, the relevant world would be actual. This means they must be mutually coherent. As possible suggests, such a world does not need to be actual. As a necessary truth, consider the proposition that for a distinct world to exist it must have particular characteristics that give it its identity. This is of course identity, which comes with correlates the classic principles of right reason. As directly rooted in such, the naturals must exist in any possible world. KF PS: I add, kindly see # 4 in this series: https://uncommondesc.wpengine.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/ kairosfocus

kf, I have no idea whether other worlds are possible, or why this world is here, so philosophical discussion of what is "necessary" don't mean much to me. You are more interested in philosophy than I am. As I've said, I take this as the only world I'll ever know, and my interest is in how it works. hazel

H, possible worlds speak is how we can put meat on the bones of necessity. The reason T is necessarily so is that in any possible world, it will obtain. This then requires that T be part of the framework for any world to exist. KF kairosfocus

Again, I agree, that countable and measureable quantities are part of our world. I know nothing about, and am not interested in, thoughts about "all possible worlds." That over my pay grade, and not something I want to consider thinking about. I'm interested in what this world is, and that's enough for me. hazel

H, crucially, major aspects of that structure are quantitative (starting with the import of identity that brings out directly 0, 1, 2 and by order type succession the naturals thus transfinite quantities also), which is why I felt it necessary to extend Neiderriter's focus on "structure" to explicitly include quantity. I thought I didn't need to explicitly expand space as well. The generality of distinct identity places the naturals etc in all possible worlds. This is the substance side. The study side is where our cultural influences and creativity come to bear. KF kairosfocus

kf, as far as I can tell, what you are saying is that our world contains structure (agreed) and that structure is such that it can be described by mathematics (agreed). (Also, complex numbers are very neat, and widely used in many important ways: agreed) Earlier I quoted with approval the Penrose quote in the OP, "Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision." (However, that quote isn't in the OP anymore??? Did wjm take that part out? If so, why? Did wjm change other parts of the OP? I'm confused???) Anyway, I've already said that there are various philosophical views on why this is all so and what it means, and that I can see validity in several different views. I don't have anything more to say about all that So, I'll stick with my summary in my first paragraph above, and leave it at that. [No change has been made to the OP- WJM] hazel

MG, if you are still hanging around. I note that the excluded middle is a corollary of distinct identity. This leads to serious problems with schemes of Mathematics that try to do away with that point. So long as items have distinct identity, a dichotomy with X-OR has come in. This is central to symbols, codes, language, including as practiced by Mathematicians. So, while one may explore limited domains which fuzz out the dichotomy (fuzzy logic with blended partial set membership comes to mind) I do not think that on the whole one can replace LEM as though it were optional. Just by using language, one is using what one would deny. KF kairosfocus

LC:

Math is a logical statement expressed in symbolic language. If the question is whether math is discovered or invented, you could ask the same thing by asking if logic itself is discovered or invented. The answer to one is the answer to the other; math can’t be discovered if logic is just an invention. Math would be an invention too. The symbology and language used to express mathematical concepts is invented . . . . if math is invented it has been invented by several species [examples were given]. I find it amazing they all reach the same results. I think the best argument is that math, like water, food, shelter and other things is part of nature and all species take advantage of them if they can.

This brings out the study-substance dual perspective I have repeatedly pointed out. In studying Mathematics we focus on structure and quantity and we use symbols, relationships and logic to guide or even frame our work. In addressing the substance in this and possible worlds, we find that distinct identity and its correlates come with the naturals, most obviously 0, 1, 2 but by way of the logic of successive order types this necessarily brings the naturals and points to the transfinites. From the naturals, we find ourselves identifying a chain of sets leading to the reals/continuum. This allows us to see how structure and quantity pervade real or virtual spaces. Space itself is structured and quantitative. Time brings in another similar aspect, and allows us to see how pervasive rates of change and accumulations of change are. (And yes, that was another topic by which I used to smuggle in advanced topics through familiar examples: water flowing into a cylindrical glass at varying rates. Yes, that included Gaussian/bell shaped impulses of change and sigmoids showing surges and plateaus of change. Those who have studied growth of organisms, movements, markets and economies will have an aha moment here.) There is much more, but enough has again been said to bring out the need to recognise the culturally influenced study AND the reality/possible worlds-embedded substance. KF kairosfocus

H, it turns out that a major part of how considerations of the substance of structure and quantity pervade science, technology and Mathematics is through the complex domain. I have already pointed to Fourier, Laplace and Z transforms. The link through these to differential and difference equations and related transfer function analyses (thus system structuring and modelling) is already enough to underscore the point. These things are pervasive in how we interact with the world and its embedded structures and quantities. Moreover, as I have repeatedly pointed out -- not merely tried to say -- such things point us to the way the world is pervaded by structure and quantity, and indeed to how much of this pervades possible worlds. All of this re-focuses the understanding that Mathematics is the [study of the] logic of structure and quantity. The use of a bracket contrasts the culturally influenced discipline that studies from the world-embedded substance that is objective and allows Mathematics to become a way that we may know certain important things about reality. And yes, this implies how truth and warrant become relevant to mathematics. Yes too, this is speaking to the interdisciplinary zone where Mathematics, Computer Science, Philosophy, Science and Technology all converge and inter-penetrate. That makes it doubly difficult but it brings out that the diverse perspectives all rightfully have seats and voices at the table. KF kairosfocus

So I've been thinking about what kf might be trying trying to point to. The world has structure and vectors are useful in describing and analyzing that structure. They are useful in the sense that they can accurately predict, within certain limits, what's going to happen as the world changes. For instance, here's a simple problem from high school physics, just involving one dimensional vectors: An object is being pulled on by two forces that are at a 60° angle. One force is 10 newtons and the other is 20 newtons. What is the net force on the object, and at what angle? So we visualize vectors, drawn as arrows, one twice as long as the other, at 60°, to represent the forces, and then use the parallelogram law and trig to solve the problem. We then measure directly the net force and angle, in some way or another, and find that our calculations were fairly accurate. Yea: our math was a good representation of the physical structure and dynamics of the situation. Of course, there aren't really little arrows attached to the object. The vectors are part of the pure math, but they do a good job of modeling the situation. The forces on the object are a real part of the structure of the situation, and the vectors accurately represent that structure. That's how I would explain it, anyway. hazel

Don't know what you are talking about, and didn't above when you wrote "study vs structure", either. But often you write sentences that I don't understand. hazel

H, chuckle again: structure. KF kairosfocus

? So what is the chuckle about? We seem to agree about the value of complex numbers as ways to represent vectors, which is a very useful and powerful part of modern science. hazel

H, I chuckle. The point of course is study vs structure. Complex numbers, particularly in real + imaginary parts form, turned out to be manifestations of vectors in the plane and useful forms for expressing same. KF kairosfocus

Again, I'm quite familiar with everything in the Wikipedia post, FWIW. But it's fun to read it. And yes, vectors have many pure and applied uses: very important tools in the math toolbox. hazel

H, sometimes the idiosyncrasies of history can make an issue more complex than otherwise. In my experience of sneaking in complex numbers a year or two early in order to better address AC theory or the like [and yes I preferred Lorentz force to the hand rules too to explain motors and generators], the rotating vector approach did not produce the reactions that I recall from 6th form Mathematics. That was a lesson. Coming back to the focal issue, it seems clear that "complex" numbers are complex as they are vectors. Indeed by historical way of Quaternions, so are ijk vectors. The quantitative structure in view then is vectors in a certain abstract logic world that captures key features that then slide over into ever so many other "pure" and "applied" contexts. KF PS: Notice Wikipedia's summary:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x^2 = -1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world . . . . The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.[4] Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.

PPS: Rearranging that exponential vector expression to give 0 = 1 + e^i*pi then reveals a key, infinitely deep structural coherence across several domains of Mathematics. I think that is part of why it is rated the most beautiful equation in Mathematics. This expression stands in handily as poster child for the logic of structure and quantity. Tendencies to down-play it are in my view symptomatic. kairosfocus

Yes, kf, I am familiar with the history of complex numbers. The invention/discovery of the complex plane as a way of representing the complex numbers as vectors was a powerful addition to mathematics, just as the number line as a representation of the real numbers was. I taught the beginnings of all this to my pre-calc and calc students. For instance, as an exercise in learning the arithmetic of complex numbers, I would have students use the quadratic formula to find the complex roots of a quadratic such as x^2 +2x +5 = 0, and then substitute one of the solutions back into the equation to demonsrate that it was indeed a solution. They were impressed that it all worked, and proud of themselves at times for being able to do it (as this was all pretty new to them.) I also showed them that connecting the endpoints of the n nth roots of a number, when plotted on the complex plane, made a regular polygon of n sides, and as a bit of a challenge had them find the three cube roots of 1 and show that they all, when cubed, did equal 1. And, of course, we learned Euler's formula for e^(ix), did some problems with it, and learned why e^(i*pi) = =-1 is just a direct result of letting x = pi. All good stuff. One of my favorite topics. hazel

H, the structure captured by the use of complex numbers is 2-d vectors in a special plane; which is a much more conceptually tractable matter than imaginary roots. And BTW "complex" strongly suggests, vectors. This approach/ perspective allows us to identify that numbers suitably represented can be 2-d quantities bearing magnitude and direction from the origin and even as rotating in time, e.g. Z = R*e^i*wt which points to the power series structure too. What came up as a way to suggest solutions to all quadratic expressions and required positing square roots of negative numbers despite the fact that [-x]^2 --> x^2, led to a gateway into a powerful domain. KF kairosfocus

Hazel, many people, and I am embarrassed to admit that I am often one of them, don’t understand the passion that some people can have for mathematics. And then I realize that I have the same passion for analytical chemistry, something that relies heavily on physics and mathematics. Ed George

Thanks, Ed. I very much enjoyed making math concepts clear, interesting, and even exciting for my students. I taught everything from general math to 9th graders to calculus (for many years) to seniors, as well as College Algebra for a local community college. I especially concentrated when I could, and especially in tech math for the non-college bound, pre-calculus and calculus, on real-world applications. hazel

Hazel, I am really enjoying your comments. You present them with such clarity that you must have been a great teacher. Ed George

That is true: math constantly grows, and often unforeseen developments improve or expand a concept. It's all cool how it works. hazel

H, yes, the numbers were introduced for algebraic reasons. My point is that often a phenomenon is run into in an odd corner then somehow we blunder our way into a wider, better picture and see things more clearly. This happened with quantum theory for example. I suspect Newton's approach to fluxions is less fruitful than infinitesimals, and they went in abeyance in turn. Then non-standard Analysis came up. KF kairosfocus

Yes, kf, using complex numbers to represent vectors was a great invention/discovery. Visualizing complex numbers on the complex plane is one of the things I like about them best. But historically this came after just using complex numbers algebraically in equations. I think this is correct, hazel

LC, Sorry but a couple of emergencies just came in over the phone so I don't have time to respond on points to your interesting suggestions just now. Please see my remarks: https://uncommondesc.wpengine.com/philosophy/logic-and-first-principles-5-the-mathemat-ical-ordering-of-reality/ KF kairosfocus

H, Pardon again, I know i was put on the table in a context which invited controversy -- indeed the sort of historical objections hinted at popped up right there in the sixth form classroom. What I pointed to was the vector phenomenon, which converts basic quantities into a plane of values. That vector approach then allows a far more natural interpretation of the exponential form and opens up the world of phasors, rotating vectors. This also points to the complex frequency domain. KF kairosfocus

I earn a living developing practical solutions to complex problems and I know my way around the mathematical block. I'm not an expert in theory. Math is a logical statement expressed in symbolic language. If the question is whether math is discovered or invented, you could ask the same thing by asking if logic itself is discovered or invented. The answer to one is the answer to the other; math can't be discovered if logic is just an invention. Math would be an invention too. The symbology and language used to express mathematical concepts is invented. If a given set of symbols is used in math, and the logic behind their use is identical to the logic of a different set of symbols, this does not make the math itself different. I don't see how anyone would disagree with me so far, but since free will does exist, I'm certain some will just because they can. And it's usually those that dispute free will, if that makes any sense. It has been claimed that since reptiles, birds and mammals all have been found to have a basic number sense and can readily see the difference between two and three, Human math ability is just built into us by evolution. So the idea that 1 + 2 = 3 does not have much of any implication as even lizards appear to know this. There is a wild Crow living near me that can count to 6 and there are accounts of other birds in other places with seemingly very canny ability to do addition and subtraction. I believe this means that math is not limited to just Human cognition. Indeed if math is invented it has been invented by several species. I find it amazing they all reach the same results. I think the best argument is that math, like water, food, shelter and other things is part of nature and all species take advantage of them if they can. LoneCycler

H, pardon but I am highlighting the subject AND substance question -- as I have emphasised throughout. On this particular issue Zeno's paradoxes of motion are an indicator of how the two must meet on equal terms. KF kairosfocus

math guy: Then how do we as purely material entities contact or perceive the platonic realm? Of course, this question asserts a fact-not-in-evidence as its premise: that we are "purely material entities." The answer to the question is "consciousness." A/mats assume that consciousness is a product of material processes. Others, such as myself, think consciousness transcends space-time and exists in the "platonic ontology." (Mathematics isn't the only realm where this sort of discussion exists. Musicians and music writers with a philosophical bent have these same sorts of dialogs. Some of us are music platonists some are not, and reasons are essentially the same. I, personally, think that all genuine creativity, contra synthetic recombination, comes from "there.") mike1962

re 112: it was originally an invention to just stand for sqrt(-1). The history of this is quite interesting, including the fact that many mathematicians rejected the concept for a long time. I believe lots of work with i as a number preceded the invention of vectors as a way of representing directed numbers. hazel

I know what calculus is, kf. :-) For many years, I enjoyed introducing beginning students to the concepts, techniques, and applications. hazel

F/N: Is "i" -- i = sqrt(-1) -- an invention or simply a notation for vectors in a particular plane of interest, the complex domain? KF kairosfocus

JAD & H: Calculus (more broadly, Analysis) is the study, rates and accumulations of flow, change, motion etc are substance and have always been with us as a quantitative, structural feature of a world where change can be discrete or continuous and variable. KF kairosfocus

Newton and Leibnitz both have some claim to having first developed calculus. Newton is usually given the most credit. hazel

I am simply underscoring the point that I made earlier @ #12 when I quoted Roger Penrose.

It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures…

https://uncommondesc.wpengine.com/intelligent-design/responding-to-ed-george-about-mathematics/#comment-669708 Who discovered (or invented) calculus? john_a_designer

PS: Lucasian chair, note Newton to Hawking. https://theconversation.com/from-newton-to-hawking-and-beyond-a-short-history-of-the-lucasian-chair-40967 kairosfocus

H, the V as palm spread out looks pretty clear to me, though hash marks -- digits as unary numerals -- is suggestive and obviously can be seen as record of a finger. Digit of course pointing to finger. Tally sticks would be a natural extension. I forgot, back in 4th form Physics, definition no 1 was: the definition of a quantity or unit is a precise statement describing that quantity or unit. KF kairosfocus

JAD, at the time physics was not really separate from Mathematics, for obvious reasons. Newton would have termed himself a natural philosopher. The Lucasian Chair is currently deemed Mathematical but has been held by Physicists. Newton's opus magnum is the Mathematical Principles of Natural Philosophy. Yes indeed to envision the fields as fluxes evenly flowing from/to a pole immediately implies inverse square law dependence as a conservation of flux principle. KF kairosfocus

Pardon typo, 5. kairosfocus

One of the most significant discoveries in science was the discovery of the inverse square law (credited to Kepler for light) which is derived directly from the geometry of a sphere. The ISL applies to both electromagnetism and gravity, though the force constants for each vary. https://www.thehighersidechatsplus.com/forums/media/inverse-square-law-and-wave-function.105/full?d=1503980290 Where would physics be without this discovery? Contrary to popular accounts Kepler and Galileo were not astronomers and Newton was not a physicist. They were all mathematicians who believed that at its root the universe was mathematical. john_a_designer

Hmmm. 5^2 - 1 = 24, which is a multiple of 24, so why wouldn't one say "every prime from 4 on [not 6] is such that on squaring will be one more than a multiple of 24"? hazel

re 100: That is a neat fact, and a nice proof. hazel

Hi math guy. First, I am a fan of complex numbers. I used to give a short history to my students about how once the decision was made to accept the square root of -1 as a number, all the rest logically followed, although it took a lot of work to figure it all out. I particular like, from an aesthetic point-of-view, the math of the nth complex roots of a number; the math of e^(ix), including Euler's identity; and the derivation of the Mandelbrot set using iterations on each point in the complex plane. However, to clarify. I didn't say prime numbers were invented: I said they were defined. Prime numbers are what they are, but someplace along the line people made a defintion (a number is prime if ...), and then started studying them. We didn't invent prime numbers themselves, but we invented (by naming and defining) the term and other symbolisms that led to us discovering the huge set of facts and properties that we are now aware of. And to kf: Wikipedia says that hash marks are a popular hypothesis about the origin of Roman numerals, but that using fingers and the palm are an alternative hypothesis. I didn't know about the palm hypothesis, but I don't think deciding which hypotheiss is correct is an important issue. I've already said, twice, that it was time for me to leave this conversation: maybe third times a charm. :-) hazel

F/N on 78: Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. Try this, every prime from 6 on is such that on squaring will be one more than a multiple of 24, cf: https://www.youtube.com/watch?v=ZMkIiFs35HQ KF kairosfocus

MG, 95: >>If we believe in the Law of Excluded Middle (which I know you do, KF) then pi is a normal number or it is not. So evidently, that Law logically implies mathematical platonism!>> Interesting observation. KF PS: It is not that I believe a list of three laws and take Aristotle as authority for centrality or even experience. I see that distinct identity is central to any particular possible world. This then directly draws out the laws, which have ontological not just cognitive import. A is itself i/l/o its core characteristics that give it its character and distinction from ROW, i.e. ~A, so W = {A|~A}. Then no x in W can be A AND ~A, LNC. Likewise any x in W will be A or else ~A, not being both or neither, i.e, the X-OR is key to LEM. We are seeing how pervasive distinct identity is and how it helps us understand so much, including numbers and extensions of numbers. kairosfocus

H, Pardon, but I find it hard to accept your response. Let me clip (noting that H has stated that she has been a Mathematics teacher), highlighting core relevant remarks: KF, 71: >> . . . The study of the logic of structure and quantity is culturally shaped but it is also constrained by the substance of structure and quantity. As has been repeatedly highlighted that starts with the import of distinct identity [--> note, I have repeatedly pointed out how this directly brings out nullity, unity, duality and extends through order types to the naturals thence the further sets]. The issue is not the cultural framing of the study, it is the necessary entity rooted substantial core. As has also been repeatedly highlighted>> ET, 72: >>hazel- inventing numerals is not inventing mathematics.>> H, 73: >>Did we discover Arabic numerals, ET.? Or perhaps you can explain what you mean by “mathematics” if you don’t include, at least in part, the symbol systems we use to write mathematics? . . . >> H, 74: >>[N]ot quite sure what your point in 71 is, kf. FWIW, the first paragraph in the longer post I am working on starts with this: “5. As kf points out, the idea of distinct identity is the starting point of math: we start with 1. From there, with the use of logic and proper definitions, we have historically built the mathematical edifice that exists today.” [--> Nope, we start with much more that is manifestly embedded in the world. The dichotomy of distinct identity W = {A|~A} directly expresses nullity, unity, duality and so successive order types with associated incremental cardinalities.] I solidly agree with you about the fundamental importance of distinct identity as the basis of mathematics: it all starts with the unit number, and builds from there.>> H, 75: >> PS to ET: I wrote, “inventing Arabic numerals and the common arithmetic algorithms we use”, not just numerals. So, to expand my question to you, do you consider the normal way we do long division, like you learned in school, part of mathematics?>> KF, 76: >>again, recall that you are speaking with people who have routinely used hexadecimals and binary digits (including binary coded decimals) so we obviously had to spend a fair deal of effort to learn different systems for numerals and even different processes such as use of twos or ones complements etc. We know the difference between the substance of structure and quantity — i.e. Mathematics — which is embedded in reality and our partly culturally shaped study of it; the discipline of Mathematics. Such is a very specific and pivotal distinction. This is clearly coming down to recognising (or refusing to recognise) that truth is the accurate description of reality. There is mathematical reality and mathematical truths use culturally conditioned symbol systems AKA codes (yes, language), to describe such then to do analysis, procedures, calculation etc.>> KF, 78: >>No, again, distinct identity PRESENTS us with nullity, unity and duality. Surely A is one thing and ~A is another; two-ness is patently present in any distinctly identifiable world . . . just as a beginning. This structure then presents us with succession from nullity, which we can analyse and term order type and thus we find the naturals. The transfinites then arise from the order type of the naturals and so forth. Something, which we were restrained from recognising by our cultural influences, even though the fact that naturals will endlessly exceed any given natural we state or symbolise say as k then exceed k+1, k+2 . . . etc as though k were only 0, is trivially demonstrable, showing a new domain of quantity, the transfinite. All of this is inherent in the rational framework of any distinct world. The work of von Neumann et al did not invent natural counting numbers. Simply the five fingers on our hands confronts us with cardinalities from 0 to 5, with the reality of equality, greater or lesser cardinalities and much more. Refusal to attend to manifest realities of the world does not transmute them into cultural artifacts just because we may study and name them, creating symbol systems to analyse, describe and use such realities. Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. However, such persistent refusal is symptomatic of cultural influences that are telling us about the state of a civilisation that increasingly throws objective truth overboard.>> ET, 77: >> I will grant only that we may have invented the symbols we use. But without mathematics this universe would not exist. Read comments 45 [Tegmark's universe made of Math thesis and discussion clipped above by me at 69] and 68 ["It is obvious that mathematics was discovered. Just ask the people who did the discovering or read what they say- start with Srinivasa Ramanujan. . . . "]>> KF, 81: >>ET, the progression from |, ||, |||, ||||, V (for the spread out palm) to 1,2,3,4, 5 is obvious and cultural. The invention of the abacus invites the place value notation and the open-bead for the empty place. Thence, power series representation on a base and the decimal point marker for whole and fractional parts. I suspect most who use decimal numbers don’t realise that they are using a compressed form for power series, where the convergence on the fractional side and the implication of endless continuation brings in the irrationals and thus the continuum. Which is another whole quantitative domain that reality confronts us with. Multidimensional continuum then allows us to see the structure and quantities of space and of shapes, lines and locations in space, including curves. Whole worlds of necessarily present properties appear. I already pointed out the use of ropes with twelve evenly spaced knots to specify a right angle. Why right angles and up/down and horizontal directions are important then comes up, embedded in properties of a terrestrial planet. At much more sophisticated level, we may STUDY how gravitation reflects warping of spacetime, and much more. The blindness to how deeply structure and quantity are embedded in reality is a sobering symptom of where our civilisation is headed now that it is hell bent on embracing inherently irrational worldviews.>> H, 79: >>I think it is important to try to make clear distinctions when discussing issues, and the points I made in 70 were meant to be about pure mathematics, not about the larger issue of mathematics in the physical world. [--> the original context of EG's challenge was a physical case, a stone falling off a cliff, but it should be obvious from ET's reference to Ramanujan that number theory is a primary reference, also if Mathematics is embedded in the world it will necessarily come out in its physics] So perhaps we can agree that the humans have invented the symbolic system that we use for representing mathematics, including symbols and definitions. [--> Notice the continued avoidance of the other half of the story, the extent of the embedded substance of Maths, starting with numbers beyond unity. I remind, distinct identity of world W leads to W = {A|~A}, thus nullity, unity and duality, two distinct entities being present in the partition and no-thing between or beyond. This already grounds the discovery of successive, endless order types per von Neumann.] Would you agree to this broader statement? >> [--> FAIR COMMENT: half the story, omitting a material part and studiously avoiding direct substantial interaction with me who has put it in play right from explaining why I find the dual character of the definition of Mathematics significant. Namely, Mathematics is [the study of] the logic of structure and quantity. The parenthesis marks the substance vs study dichotomy, the substance being in the world and discovered, the study engaging the substance and using culturally influenced symbols, axiomatisations, codes, procedures etc. This should not have to be belaboured but that is where we are.] H, 80: >>kf, to what does “no, again” refer in 78? Or does your post have anything to do with what I have written? [--> subtext of evasion]>> ET, 82: >>we may have invented the symbols. That is all I will grant. What did Ramanujan discover? Was it the equations fully equipped with the symbols he didn’t know? Or did he project the known symbols onto the equations he was given? For me I would say that we have discovered it all. That is because all of the information has been loaded into this universe.>> KF, 83: >>H, I spoke to your recent comments, and have made sufficiently specific remarks that that may be readily discerned. [--> as in study vs substance, distinct identity directly indicating 0, 1, 2 and thence successive order types embracing the Naturals etc. Notice also, how often I have used the extension to reals and continuum to highlight the structure-quantity aspects of space, even in an earlier thread speaking to moving fingers across space to type comments]>> H, 85: >> in 78, kf wrote, Another capital example [of what I am not sure] is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. I think if you were to re-read 70, you would see that I would agree with you that, once prime number is defined, “their many fascinating properties and patterns” were discovered, not invented. We are in agreement on that issue.>> [--> further studious evasion of substance vs subject and the extent to which structure and quantity are embedded in a spatio-temporal world in which we type comments by moving fingers and rocks fall off cliffs at rates of change of displacement exhibiting rates of change of such velocity as are manifest in acceleration tied to force-inertia balances] H, 88: >> Well, I didn’t quit. I think I understand what kf is trying to say. [--> notice condescending tone, I am not merely inchoately attempting like a spermologos picking up ill-digested scraps around the Agora] I have been talking about pure mathematics, and I agreed that recognizing the important of distinct identity, the unit one, is the foundation upon which all mathematics is built. [--> Notice the repeated studious evasion of the inconvenient 0, 1, 2 and implied onward succession to N in W = {A|~A}] His point is that we experience distinct identity in the world: five fingers, or a piles of stones. This is true, and is the underlying experience which motivated the very beginning of pure mathematics. [--> Notice, again, evasion of substance vs study] The history of this is very interesting, and I have written about it in the past (although I don’t know if I’ve done that here.) Before a written system was developed, there is anthropological evidence that people used a one-to-one correspondence between a group of pebbles and some other group, such as a herd of sheep, to compare quantities. [--> thus, equality is embedded in reality, manifest in matching set cardinalities] Eventually simple hash marks substituted for the pebbles – the first written numbers were simply vertical hash marks with a slash through every four to represent a group of five. [--> Tally-marks are not to be equated to original form Roman Numerals which use the V to show the span of five fingers] If [--> the loaded if] pointing to this is what kf is trying to do [--> In this context I explicitly highlighted Roman Numerals, as they are the most blatant case in point of using cardinalities manifest in our fingers and toes] , then I agree. We started the building of pure mathematics by inventing ways to write about the simple distinct identities and quantities that we observed in the world around us. [--> again, evades discussing the import that much more than 1 is embedded in the world, i.e. our world is imbued with, pervaded by manifest, multiple forms of structure and quantity, manifesting the core substance of Mathematics]>> H, 90: >>Taking ET seriously [--> by contrast with . . . ], pure mathematics is mathematics done within a logical system, without reference to any application or reference to things outside the system. [--> Oh, what was all of that silly stuff on abstract logic-model worlds that are poossible worlds that can connect to other worlds through family resemblance or even through discovery of necessary entities framework to all worlds? Oh, nothing worth noticing . . . ] For instance, I think the vast majority, if not all, of the work done by Ramanujan, whom you referenced, was in pure mathematics. Did you scroll through the Wikipedia article and see some of them: that’s pure mathematics. Some of his discoveries may have found later application (I don’t know if that is true or not), but my informed guess that most of them have no relevance to anything outside pure mathematics. >> [--> R's work was a materially independent discovery of and in parts extension to number theory, which brings out properties of the set N which draws out from distinct identity through 0,1,2 and order type succession, thus is a transfinite set of necessary beings embedded in the framework of any possible world] ET, 91: >>Discoveries is the operative word, hazel.>> H, 92: >>Yes, I know. Re-read post 70. I think I’ve been clear about what part of pure mathematics is invented and which part discovered. >> [--> Just the opposite, H suppressed the extent and import of what is discovered. Not to mention, this evades conceding explicitly that the distinction between embedded locked in the world substance and culturally influenced study that gains objectivity by having to correspond to that substance is fully justified.] The rhetorical patterns and the balance on substantial merits now stands clearly revealed through this evasive half-concession. KF kairosfocus

MG, of course, while I studied Maths itself, much of my interest was on the side of systems, where the complex frequency domain in its various guises is a huge domain of praxis. Thus i (or j) becomes a key to seeing the frequency and transient structure of systems behaviour thanks to Laplace and Fourier. Spotting poles and zeroes can be a bit of a sport, whether you think s-domain or the linked z-domain. The latter of course being particularly suited to discrete-state systems. This brings out how a logic model world (ultimately underpinned by relevant axioms, postulates etc) can function as a possible albeit abstract world that then highlights things that may be necessary entities present in all worlds or just things present in worlds in the neighbourhood or family of the one we live in. But underlying, we see that once inertia, dynamics, energy storage, friction and the like or equivalent are present so differential equations or difference equations capture behaviour, just the structure of the equations reveals a lot about transient and frequency-oriented behaviour. Again, substance of structure and quantity are pivotal. I also tend to use j as an orthogonal rotation operator which then naturally brings out how sqrt-(-1) is j: j*x --> y, j*y --> -x, i.e. -1*x so j = sqrt-(-1). This then naturally brings in rotation, phasors and frequency. So, complex numbers are actually a way to do vectors algebraically, as the extension to the ijk basis further illustrates. KF kairosfocus

math guy @ 93,

Were the complex numbers (the field C, not the symbolic representations x+iy) non-existent until Tartaglia “invented” i, after which they sprang fully formed into existence like Athena out of Zeus’ head? Now a very interesting thing about these complex numbers is their relation to the distribution of prime numbers. The Riemann Hypothesis says that a certain regularity result about the distribution of primes is equivalent to a complicated complex-valued function of a complex variable having its roots lie solely on the line 1/2+yi. Why should Tartaglia’s “invention” dictate behavior of prime numbers?

Or, if we invented math, does that mean we invented irrational numbers like pi and the square root of 2? Apparently it was the Pythagoreans who “discovered” irrational or alogos numbers. In fact, the discovery was credited to a guy by the name of Hippasus, though according to legend they were not too happy about his discovery and while out at sea, they threw him overboard. john_a_designer

KF, Regarding the decimal expansion of pi, another famous conjecture is that pi is a "normal" number. This means that taking any sufficiently large initial segment of its decimal expansion, the digit "7" will occupy 1/10 of the segment, the string "48" will occupy 1/100 of segment, and so on. In particular any finite string of n integers will appear in the expansion, and at proportion 10^{-n} for sufficiently large initial segments of the expansion of pi. If we believe in the Law of Excluded Middle (which I know you do, KF) then pi is a normal number or it is not. So evidently, that Law logically implies mathematical platonism! math guy

WJM has hypothesized that mathematical symbols are what we humans invent. Fields Medalist (i.e. really smart guy) Allain Connes suggests that humans invent formal systems, or at least the axioms thereof, as the tools used to investigate what he calls primordial mathematics (the relationships and structures that seem inherent to platonists). A-mats claim to debunk platonism as follows (idea first espoused by Quine): suppose there is a platonic realm containing all of primordial mathematics (and perhaps lots more). It is of course immaterial and not part of the universe we occupy, since everything here is only imperfect shadows of the platonic realm which decay and die. Then how do we as purely material entities contact or perceive the platonic realm? I see primordial mathematics as a striking counterexample to materialism. Since I myself DO get occasional glimpses of the platonic realm of primordial mathematics, I am convinced it is real and therefore materialism cannot be true. My subjective experience is evidence (to me) that humans are more than machines made of meat; that we do have some non-physical contact with an eternal unchanging reality. math guy

Another phenomenon (already hinted at by hazel) is that "inventing" some mathematical structure (she mentions prime numbers) invariably leads to all sorts of unforseen consequences and subsidiary facts about the structure, only established by mathematical proofs. A classic example is the imaginary unit i, first postulated as the "imaginary unit" by Tartaglia, Cardano, et al. This was a crutch used to find roots of cubic and quartic polynomials. What about higher order polynomials? Do we need new "imaginary" numbers to find their roots? No, Gauss, et al proved that the complex field C is algebraicly closed, meaning that the roots of any polynomial with coefficients in C also lie in C = R(i). Were the complex numbers (the field C, not the symbolic representations x+iy) non-existent until Tartaglia "invented" i, after which they sprang fully formed into existence like Athena out of Zeus' head? Now a very interesting thing about these complex numbers is their relation to the distribution of prime numbers. The Riemann Hypothesis says that a certain regularity result about the distribution of primes is equivalent to a complicated complex-valued function of a complex variable having its roots lie solely on the line 1/2+yi. Why should Tartaglia's "invention" dictate behavior of prime numbers? math guy

Yes, I know. Re-read post 70. I think I've been clear about what part of pure mathematics is invented and which part discovered. hazel

Discoveries is the operative word, hazel. ET

Taking ET seriously, pure mathematics is mathematics done within a logical system, without reference to any application or reference to things outside the system. For instance, I think the vast majority, if not all, of the work done by Ramanujan, whom you referenced, was in pure mathematics. Did you scroll through the Wikipedia article and see some of them: that's pure mathematics. Some of his discoveries may have found later application (I don't know if that is true or not), but my informed guess that most of them have no relevance to anything outside pure mathematics. hazel

"Pure mathematics"? Is that what you get when you run mathematics through a series of filters? :cool: ET

Well, I didn't quit. I think I understand what kf is trying to say. I have been talking about pure mathematics, and I agreed that recognizing the important of distinct identity, the unit one, is the foundation upon which all mathematics is built. His point is that we experience distinct identity in the world: five fingers, or a piles of stones. This is true, and is the underlying experience which motivated the very beginning of pure mathematics. The history of this is very interesting, and I have written about it in the past (although I don't know if I've done that here.) Before a written system was developed, there is anthropological evidence that people used a one-to-one correspondence between a group of pebbles and some other group, such as a herd of sheep, to compare quantities. Eventually simple hash marks substituted for the pebbles - the first written numbers were simply vertical hash marks with a slash through every four to represent a group of five. If pointing to this is what kf is trying to do, then I agree. We started the building of pure mathematics by inventing ways to write about the simple distinct identities and quantities that we observed in the world around us. hazel

kf, I don't "readily discern" at all how what you wrote responds to the specific things I've written. But I will quit trying if you can say no more. hazel

As far as whether or not any particular person "finds the arguments compelling", what possible difference does that make? One either has a rational criticism based on fact, evidence and/or logic of the arguments presented, or they do not. One either has a rational argument for the idea that humans invented mathematics and did not discover it, or they do not. William J Murray

in 78, kf wrote,

Another capital example [of what I am not sure] is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented.

I think if you were to re-read 70, you would see that I would agree with you that, once prime number is defined, "their many fascinating properties and patterns" were discovered, not invented. We are in agreement on that issue. hazel

Ed George said:

William felt that my argument was sufficient to warrant a dedicated OP. Maybe you should take it up with him.

Incorrect. As far as I can tell, you've made no argument for your position. Perhaps that is what hazel is attempting to do with that person's latest posts, I don't know. What I felt was that your comments and criticisms were based on a failure to understand the argument being made in favor of what mathematics represents being a discoverable, inherent aspect of the universe. My post was about presenting that argument in a different light in order to try and provide understanding. So far, I haven't seen anyone answer or address this very simple question pertaining to whether we invent or discover mathematics: Did humans discover that 2+3=5? Or did they invent it? I’m not talking about the symbols; I’m talking about what the symbols refer to. For example, did we just invent the fact that 2 of anything plus 3 of the same thing equals five of that thing? William J Murray

H, I spoke to your recent comments, and have made sufficiently specific remarks that that may be readily discerned. KF kairosfocus

hazel, we may have invented the symbols. That is all I will grant. What did Ramanujan discover? Was it the equations fully equipped with the symbols he didn't know? Or did he project the known symbols onto the equations he was given? For me I would say that we have discovered it all. That is because all of the information has been loaded into this universe. Starlings, held in captivity never seeing the night sky, become restless and wanting to migrate when shown the night sky that would trigger their normal migration. "Instinct" is juts a word for "we have no idea". For me they are tapping into the Starling information vault. ("See "Why is a Fly Not a Horse?") ET

ET, the progression from |, ||, |||, ||||, V (for the spread out palm) to 1,2,3,4, 5 is obvious and cultural. The invention of the abacus invites the place value notation and the open-bead for the empty place. Thence, power series representation on a base and the decimal point marker for whole and fractional parts. I suspect most who use decimal numbers don't realise that they are using a compressed form for power series, where the convergence on the fractional side and the implication of endless continuation brings in the irrationals and thus the continuum. Which is another whole quantitative domain that reality confronts us with. Multidimensional continuum then allows us to see the structure and quantities of space and of shapes, lines and locations in space, including curves. Whole worlds of necessarily present properties appear. I already pointed out the use of ropes with twelve evenly spaced knots to specify a right angle. Why right angles and up/down and horizontal directions are important then comes up, embedded in properties of a terrestrial planet. At much more sophisticated level, we may STUDY how gravitation reflects warping of spacetime, and much more. The blindness to how deeply structure and quantity are embedded in reality is a sobering symptom of where our civilisation is headed now that it is hell bent on embracing inherently irrational worldviews. KF kairosfocus

kf, to what does "no, again" refer in 78? Or does your post have anything to do with what I have written? hazel

Thanks, ET. I think it is important to try to make clear distinctions when discussing issues, and the points I made in 70 were meant to be about pure mathematics, not about the larger issue of mathematics in the physical world. So perhaps we can agree that the humans have invented the symbolic system that we use for representing mathematics, including symbols and definitions. Would you agree to this broader statement? hazel

PS: No, again, distinct identity PRESENTS us with nullity, unity and duality. Surely A is one thing and ~A is another; two-ness is patently present in any distinctly identifiable world . . . just as a beginning. This structure then presents us with succession from nullity, which we can analyse and term order type and thus we find the naturals. The transfinites then arise from the order type of the naturals and so forth. Something, which we were restrained from recognising by our cultural influences, even though the fact that naturals will endlessly exceed any given natural we state or symbolise say as k then exceed k+1, k+2 . . . etc as though k were only 0, is trivially demonstrable, showing a new domain of quantity, the transfinite. All of this is inherent in the rational framework of any distinct world. The work of von Neumann et al did not invent natural counting numbers. Simply the five fingers on our hands confronts us with cardinalities from 0 to 5, with the reality of equality, greater or lesser cardinalities and much more. Refusal to attend to manifest realities of the world does not transmute them into cultural artifacts just because we may study and name them, creating symbol systems to analyse, describe and use such realities. Another capital example is the reality of primes and their many fascinating properties and patterns, which, manifestly, we discovered rather than invented. However, such persistent refusal is symptomatic of cultural influences that are telling us about the state of a civilisation that increasingly throws objective truth overboard. kairosfocus

hazel, I will grant only that we may have invented the symbols we use. But without mathematics this universe would not exist. Read comments 45 and 68 ET

H, again, recall that you are speaking with people who have routinely used hexadecimals and binary digits (including binary coded decimals) so we obviously had to spend a fair deal of effort to learn different systems for numerals and even different processes such as use of twos or ones complements etc. We know the difference between the substance of structure and quantity -- i.e. Mathematics -- which is embedded in reality and our partly culturally shaped study of it; the discipline of Mathematics. Such is a very specific and pivotal distinction. This is clearly coming down to recognising (or refusing to recognise) that truth is the accurate description of reality. There is mathematical reality and mathematical truths use culturally conditioned symbol systems AKA codes (yes, language), to describe such then to do analysis, procedures, calculation etc. KF kairosfocus

PS to ET: I wrote, "inventing Arabic numerals and the common arithmetic algorithms we use", not just numerals. So, to expand my question to you, do you consider the normal way we do long division, like you learned in school, part of mathematics? hazel

ot quite sure what your point in 71 is, kf. FWIW, the first paragraph in the longer post I am working on starts with this: "5. As kf points out, the idea of distinct identity is the starting point of math: we start with 1. From there, with the use of logic and proper definitions, we have historically built the mathematical edifice that exists today." I solidly agree with you about the fundamental importance of distinct identity as the basis of mathematics: it all starts with the unit number, and builds from there. So, I'm curious what your thoughts are on the points I made at 70? Is there anything there you disagree with? hazel

Did we discover Arabic numerals, ET.? Or perhaps you can explain what you mean by "mathematics" if you don't include, at least in part, the symbol systems we use to write mathematics? Also, are there other parts of my post at 70 that you agree or disagree with? I'm curious to know some of what you think about this part of the topic. hazel

hazel- inventing numerals is not inventing mathematics. ET

H, you are talking with people who routinely have worked in hexadecimals and binary. The study of the logic of structure and quantity is culturally shaped but it is also constrained by the substance of structure and quantity. As has been repeatedly highlighted that starts with the import of distinct identity. The issue is not the cultural framing of the study, it is the necessary entity rooted substantial core. As has also been repeatedly highlighted. KF PS: You don't want to get me going on the Alinskyite rhetorical habit of red herrings led away to strawman caricatures soaked in ad homs and set alight to cloud, confuse, poison and polarise the atmosphere. This is a discussion on mathematics which of all things should be in-common. That it is manifestly not so is yet another indicator of the state of our civilisation and especially its formal and informal education systems. kairosfocus

ET, some examples of math that have been invented are Arabic numerals and the common arithmetic algorithms we use, coordinate geometry, and the f' notation for derivatives. A common distinction is that mankind has invented the particular symbol systems that we use, but within those systems, once established, the logical consequences are then discovered as inevitable logical consequences. For instance, once the number system was extended to include complex numbers, Euler's Identity e^(i*pi)= -1 was discovered. And Euler's identity is a true fact irrespective of what symbol system is used (we might have used different symbols of i and pi, or a different way of indicating exponentiation, for instance), and it was just as true before we discovered it (just unknown) as it is now that we have discovered it. Are there any parts of what I written here that you agree with, and if so, which. Are there any parts you disagree with, and if so, which, and why> hazel

H'mm: Without endorsing his full thesis (some sort of global math sim run as part of a quasi-infinite multiverse so far as I can see . . .), let's clip Tegmark:

Equations aren't the only hints of mathematics that are built into nature: there are also numbers. As opposed to human creations like the page numbers in this book, I'm now talking about numbers that are basic properties of our physical reality. For example, how many pencils can you arrange so that they're all perpendicular (at 90 degrees) to each other? 3 – by placing them along the 3 edges emanating from a corner of your room, say. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there 3 dimensions rather than 4 or 2 or 42? [--> note, embedding of structure and quantity, simplified form] And why are there, as far as we can tell, exactly 6 kinds of quarks in our Universe? There are also numbers encoded in nature that require decimals to write out – for example, the proton about 1836.15267 times heavier than the electron. From just 32 such numbers, we physicists can in principle compute every other physical constant ever measured. There's something very mathematical about our Universe, and that the more carefully we look, the more math we seem to find. So what do we make of all these hints of mathematics in our physical world?

Mathematics, substantially is the logic of structure and quantity. Where, logic speaks to rational framework of principles. Some of that substance as I showed is inherent in there being a possible world W, i.e. that necessarily involves W = {A|~A} thus nullity, unity, duality thence by successive order types the naturals and even transfinites. From this, we go to rationals, irrationals, Reals thus continuum. This shows systematic structure and quantity embedded in space. That is already sufficient. At this point, I think there is something at worldview level that leads to the sort of resistance we have seen, but so far no substantial case has been made to explain it, at least from objectors. I am inclined to point to culturally diffused Kantian thought locked to cultural relativism and subjectivism. That is, the perceived ugly gulch between the phenomenal world of appearance (individual or collective) and the noumenal one of things in themselves, held to be inaccessible as all of our senses and reflective capabilities are fatally warped. From F H Bradley on, this has been shown to be self-referentially incoherent. In effect, to so claim to deny knowledge of the outer world is to make just such a knowledge claim. Thus, self-defeat. Though, such a perception may easily be self-reinforcing especially if it fits strongly reinforced cultural agendas. The pivotal centrality of the principle of distinct identity and of its linked import that natural numbers, quantities more broadly and ordering structures (math sense not just spatial . . . I am not committing to worlds being spatially extended) necessarily pervade any possible world. Mathematics is effective in science because it is part of the framework for a possible world thus the actual one. And no it is not just internal manipulations and subjective meanings -- what "semantics" typically means when it is used dismissively. This is objective, locked into the framework for a possible world. As for oh we invented mathematics, we do pursue it in accord with cultural traditions, that's the study part. Those vary but are constrained by the substance part as seen, Math facts on steroids embedded in reality in effect. This interlock is why we may freely make up abstract, axiomatised logic-model worlds then use them to explore consequences; creating possible worlds. Such worlds will have necessary, framework structural components tied to distinct identity. They may also have contingent aspects that are useful in setting up systems close enough to our world to be useful without being necessarily true or even contingently true. It is perfectly possible in logic that a false antecedent implies true consequents, just that we have to be careful over range of valid application. We should not let the contingent part distract us from the necessary part. KF kairosfocus

It is obvious that mathematics was discovered. Just ask the people who did the discovering or read what they say- start with Srinivasa Ramanujan. From there read the book that I linked to in comment 45- "Our Mathematical Universe". And it should be noted that no one has put forth any argument that humans invented mathematics. ET

JaD

Does he have an argument backed with reason and evidence? Or does he believe that his personal opinion settles the issue?

William felt that my argument was sufficient to warrant a dedicated OP. Maybe you should take it up with him. [Posts that are entirely about personalities and no substance will be deleted - WJM] Mathematics is either inherent in the universe and was discovered by man, or it was invented by man to model the universe. Other than as a long standing philosophical question, the answer to the question has no importance to our lives. Based on the diversity of opinions here, I am not the only one who does not find the arguments put forward in support of mathematics being inherent to the universe very compelling. If someone presents a compelling argument, I will gladly change my opinion. Ed George

What is Ed George’s reason for being here? As far as I have seen he has as yet to make an argument. The interaction all started with EG and a couple of other interlocutors on the “Logic & First Principles, 4” thread on 12/11. Here is a brief summary. Pater Kimbridge @ #4:

One must be careful not to slide into a pit of equivocation here. The universe has structure and quantity, but numbers are an invention of Man. In fact, all of mathematics is an invention of Man.

https://uncommondesc.wpengine.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/#comment-669555 Belfast @ #6 concurred:

PK is correct. Nature knows nothing of anything bigger than 1. Two atoms are different from each other in location at least. Mathematics invented by man enables man to quantify order and states and understand patterns at various levels.

Ed George the weighed in @ #11: “For what it’s worth, I have to side with PK on this.” @ #13 KF who wrote the OP then challenged him: ”EG, make your case: _______ KF” EG responded @ #17: “It is my opinion that mathematics is a human invention that can be used to model the world that we see around us. For example e = mc^2 means absolutely nothing without first defining energy, mass and the speed of light.” @ #19 I weighed in:

So what is EG’s argument? Either X or Y could be true EG believes Y Therefore, Y is true. In other words, Ed George believes it. That settles it.

EG responded @ #21:

That is all any of us can do. Mathematics either exists independent of humans or it is something invented by humans to model our observations. My opinion is that it is the latter. ET and KF believe it is the former. But, unfortunately, there is no way of determining which is true. And, frankly, does it matter?

Since then several of us have given reasons for our position. Again, as far as I have seen EG has not. Does he have an argument backed with reason and evidence? Or does he believe that his personal opinion settles the issue? I don’t see why he continues to hang around. He has, however, however succeeded in being disruptive. Maybe that’s his reason. john_a_designer

I have started what will be a long post summarizing the issues here as I see them. I have written the following as an introduction which I might as well post now, with the understanding that the rest is forthcoming: mathguy has joined the conversation, so I'm going to try again. 1. I am definitely not Ed George, and don't know the degree which we might agree on things, as I hadn't read the posts of his in the previous threads that prompted the OP by wjm. He responded positively to one thing I wrote, and don't think I've responded to any of his posts. So let's keep me separate from him, please. 2. I looked back over my posts in this thread: 11, 17, 19, 29, 33, 34, 39, 42, and 50, which is the one where I pointed out that what I felt was unwarranted hostility didn't make carrying on the conversation much fun. I don't think characterizing this as "slinking away" is very useful. Perhaps the tone of the conversation can improve. 3. Also, I don't think "defeated in debate" is an accurate characterization. I am not debating, I am trying to discuss. This is a perennial philosophical issue, and as I said in an earlier post, I find that I can slide back and forth between different views, much like the famous Gestalt picture of the old/young lady. So, if you look at my previous posts here, you'll see that I've asked questions, tried to explain the points I'm not clear in my mind about, pointed out points I agree on, and asked if I was correctly understanding other's points. This is what people do when they discuss, not when they debate. This is also why I didn't understand the hostility I felt in some responses back to me. 4. Just a bit of background. I have been a math teacher for many years. I have read fairly widely about the philosophy of math, and am familiar with issues like infinite series, the Mandelbrot set, Euler's identity e^(i•pi) = -1, etc. I'm not claiming expertise here, but rather pointing out that I, too, have a long interest in this subject and enough background to discuss it knowledgeably. With all those disclaimers, I'm going to try to summarize the issue as I see it, identifying specific points so that maybe we can be clear on what aspects of the situation we are in agreement on, and which we might not be. I welcome people pointing out areas of agreement, disagreement, shared questions, etc. ... I hope to finish the rest of the post later today, but I'm interested in taking the time to be fairly thorough. hazel

Math Guy

I think hazel and Ed George are the same person with two accounts. This person, defeated in debate, has slunk away.

Sorry, but Hazel and myself are different people. And I can assure you that I have not skunk away. I have read all of the comments. I just haven’t seen anything that is compelling enough to respond to. Ed George

LC, the interesting thing is why people resist that. KF kairosfocus

I don't think its that difficult to see the mathematical nature of the World. Just the opposite; you have to work very hard to ignore it. Fractals are one example. Someone please explain why the shape of a river, a tree and an ice crystal would be similar. To me it seems the designer put these things here for us to notice, to investigate, to learn. Of course, some won't. https://www.mnn.com/earth-matters/wilderness-resources/blogs/14-amazing-fractals-found-in-nature A-mats will claim these aren't "real" fractals, because they don't occupy infinite levels as pure matehmatical fractals do. LoneCycler

Exploring a million digits of pi: https://uncommondesc.wpengine.com/intelligent-design/exploring-a-million-digits-of-pi-a-vid/ kairosfocus

PS: Just went looking for a million decimal digits of pi. https://www.piday.org/million/ four-strings of 0's and 7's are present on using the find in page tool. That supports the conjectures on longer strings of same. Again, an inductive argument in a Mathematical context. Trillion digit out of 22.4 download here: https://pi2e.ch/blog/2017/03/10/pi-digits-download/ kairosfocus

MG, thanks for the interest and for un-lurking. You have raised a significant point, that the bulk of Mathematicians believe that mathematical entities and linked states of affairs are realities such that their actuality and truthful (or false) description is a meaningful discussion. The specific cases are significant and remind me of the debate over the proof of the four colour map theorem about forty years back. On pi, I have taken large expansions as random number tables and accept on experience that for a large enough expansion essentially any string of digits will be there. I have found phone numbers for example. But while I ponder whether a string of 0's or of 7's is there, I believe that the reality exists independent of my knowledge or our knowledge, which then allows us to describe and recognise truth as accurate representation of actuality. From my perspective, we can see many key things for a simple but significant case: WLOG, for a distinct world Wk to exist, the dichotomy tied to its identity must obtain so we see Wk = {A|~A}, which brings with it the chain 0, 1, 2. Succession of order types then indicates the naturals and transfinites. These are necessary entities, part of the framework for any given possible world to exist. This opens up the logic of being question, of what nature is a natural number or a transfinite then onward the panoply of rationals, integers, irrationals that give us the reals and the continuum. Which cannot be severed from the experience of extended space. One thing I see there is how the interval [0,1] has in it the images for 1/x, x GRT 1, so we see how transfinite cardinality is not like that of finite sets. Now, numbers and other abstracta are informational and it seems, mental. So, part of the intractability of the debates will be the unwelcome shadow of God on the doorstep of the temples of the radically secularist academy. Where of course, here, God is a serious candidate necessary being. One who would contemplate eternally the panoply of number, quantity, structure etc that is the essence of Mathematics. But, such would be an onward question. What is of interest here is that Mathematics interacts with possible worlds discourse in highly interesting ways. So, when we use [culturally influenced!] axiomatisations to develop abstract logic-model worlds, we are exploring possibilities, or at least provisional ones [nod to Godel]. If in so doing we stumble upon necessary beings, these would be part of the framework for any possible world, thus our actual one. And this answers Wheeler's itch. Mathematics is powerful in the physical sciences as it contains necessary being entities that are integral to the framework for any world that is possible of existence. Whether physical or as a computer simulation or as a mental contemplation makes but little difference. In turn, this encourages exploration and identifies strategic targets. Going further, the concept of classes or achetypes that are possible beings in possible worlds with clusters of core characteristics extensible to objects instantiating the classes then becomes significant. As I argued in the first few OP's in my current series, this grounds reasoning by analogy and by induction more generally. As, if characteristics c1 . . . cm are shared by all members or at least this is credible then to see sufficient family resemblance supports conclusions that turn on such inheritance. All of this invites exploration. KF kairosfocus

EG, I have not personally insulted you; that is neither my general habit nor do I hold animosity to you. I suggest that you turn down the rhetorical voltage and deal with substantial issues. KF kairosfocus

A more mature but common example is seen all the time in my profession. We pose open problems referred to as "conjectures". Two famous conjectures were solved relatively recently: Fermat's Last Theorem and the Poincare Conjecture. Were they "meaningless" prior to their solution? Of course not! Almost everyone believed they were true and worked hard for many years (300+ years for FLT) to find a proof. That's platonism in action. (Fame and fortune for correct solutions also helped.) There is another million dollar problem (Clay Institute will pay for a valid solution) called the Navier-Stokes Equation. This is a mathematical model for turbulence that has important applications in the physical world. Is there or is there not a tractable solution for NS? That depends on whether you are a platonist! math guy

I've been lurking for a number of years, but this OP is a particular interest of mine and I could not resist responding! Disclosure, I am a mathematical platonist who gets frustrated at the obstinacy of those who deny what is patently obvious (to me and most mathematicians). In the style of Granville Sewell, let's think about what a non-platonist accepts: Does the decimal expansion of pi contain 100 consecutive occurrences of the digit "7"? Any high school student would respond to the effect, it does or it doesn't, we just don't know yet. That's platonism. The non-platonist will have to quote AJ Ayers and claim the question is meaningless. But what does Ayers say after the latest supercomputer churns out another trillion digits for pi that contains the requisite sequence of "7"s? That we just created that sequence from nothing? Our original question now has meaning? math guy

[Posts that are entirely about personalities and no substance will be deleted - WJM] math guy

[Posts that are entirely about personalities and no substance will be deleted - WJM] Ed George

EG, the issues remain on the table. For days, they have stood essentially unanswered from your side. Do you think that that will not be noticed leading to the conclusion that the objections lack substance? KF kairosfocus

Hazel

Hmmm . You guys are awfully hostilely argumentative, and I don’t know where the hostility comes from.

I have to agree. All we are debating is whether mathematics is something that humans invented to model (overlay) reality, or whether it existed, somehow (but never explained) before humans arrived. Personally, I am fine with either, but I just don’t find the arguments for mathematics being fundamental to existance being very compelling. Ed George

H'mm: H, 44: >>A circle is an abstract concept. No circle, in the mathematical sense, exists in the physical world. Do you agree, ET?>> ET or other personalities not being primary, I responded, inviting an examination on the merits: KF, 46: >>Hazel, We can worry about the precise ontology and metaphysics of numbers, the continuum, motion involving rates of change tied to displacement, etc later. Right now, start from, is motion a real phenomenon? If not, explain how you typed your comments through the delusion of motion. If so, rates of change and connected locations in space across time are sufficient to underscore the point.>> EG, 47: >>Hazel @ 44, the same applies to right angles, straight lines and cubes. We can describe them mathematically, but they don’t really exist in nature. We can use math to model acceleration due to gravity, but at best we are talking about an average. Acceleration due to gravity on the bar stool I am currently sitting is different than it is on the stool at the other end of the bar.>> KF, 48: >>Ed George kindly describe nature in such a way that it does not include space. For, if nature manifests space, right angles — a spatial phenomenon — most certainly exist. Namely any two vectors such that they start at the same point and are orthogonal so the cos of their included angle is zero. Of course, you will be tempted to deny points, which simply denotes location. If what you mean is that we cannot make an absolutely perfectly right angled body, that is both irrelevant and points to the real problem being the assumption that “real” means physical-material. That would show the pernicious influence of evolutionary materialistic scientism, which is self-refuting and necessarily false. But then, my saying that brings up propositions (the truth claims asserted or denied in statements) which strict materialists deny as being real. The problem then being that meaning, logical implication, logical antithesis and distinct identity also have not mass nor location nor size nor energy content and would become unreal, again illustrating the absurdities of such materialism and of fellow traveller movements>> WJM, 49: >>hazel, I have certainly not made the argument that abstractions or “Platonic forms” exist in what we call the real world or as an overlay. I demonstrated that the abstractions are symbols of language used to describe real things we find in the world, and showed how the word “tree” is an abstraction that points to a real thing, and how mathematics are abstractions that point to real thing (behaviors), and so saying “we invented mathematics” instead of “we discovered mathematics” is exactly the same as saying “we invented trees” or “we invented gravity” or “we invented inertia”. This is obvious, basic logic. So you moved the goal posts. Instead of actual things denoted by symbolic words, you moved to a set of characteristic like “tree-ness” or “sphere-ness”. I also addressed that directly. If “sphere-ness” is comparable to “mathematics-ness” and used to represent qualities found in physical phenomena, then once again it is appropriate to consider mathematics a discoverable, not invented, phenomena. Humans did not invent spherical objects, nor did humans invent the mathematical behaviors of phenomena. Now you’ve moved the goal post yet again, to claim the debate is about whether not “platonic forms” are part of the real world. And, you and Ed, apparently in lieu of any actual rebuttal of these points, invoke a quote from Einstein as if that is a meaningful addition to the debate here. How about answering a simple question: did humans discover that 2+3=5? Or did they invent it? I’m not talking about the symbols; I’m talking about what the symbols refer to. For example, did we just invent the fact that 2 of anything plus 3 of the same thing equals five of that thing? >> H, 50: >>Hmmm . You guys are awfully hostilely argumentative, and I don’t know where the hostility comes from. But it’s not much fun, so I’ll bow out.>> It seems, there are some serious points on the table that need an answer from those who wish to argue that the substance of Mathematics is not embedded in the world. KF kairosfocus

[Posts that are entirely about personalities and no substance will be deleted - WJM] hazel

hazel, I have certainly not made the argument that abstractions or "Platonic forms" exist in what we call the real world or as an overlay. I demonstrated that the abstractions are symbols of language used to describe real things we find in the world, and showed how the word "tree" is an abstraction that points to a real thing, and how mathematics are abstractions that point to real thing (behaviors), and so saying "we invented mathematics" instead of "we discovered mathematics" is exactly the same as saying "we invented trees" or "we invented gravity" or "we invented inertia". This is obvious, basic logic. So you moved the goal posts. Instead of actual things denoted by symbolic words, you moved to a set of characteristic like "tree-ness" or "sphere-ness". I also addressed that directly. If "sphere-ness" is comparable to "mathematics-ness" and used to represent qualities found in physical phenomena, then once again it is appropriate to consider mathematics a discoverable, not invented, phenomena. Humans did not invent spherical objects, nor did humans invent the mathematical behaviors of phenomena. Now you've moved the goal post yet again, to claim the debate is about whether not "platonic forms" are part of the real world. And, you and Ed, apparently in lieu of any actual rebuttal of these points, invoke a quote from Einstein as if that is a meaningful addition to the debate here. How about answering a simple question: did humans discover that 2+3=5? Or did they invent it? I'm not talking about the symbols; I'm talking about what the symbols refer to. For example, did we just invent the fact that 2 of anything plus 3 of the same thing equals five of that thing? William J Murray

Ed George kindly describe nature in such a way that it does not include space. For, if nature manifests space, right angles -- a spatial phenomenon -- most certainly exist. Namely any two vectors such that they start at the same point and are orthogonal so the cos of their included angle is zero. Of course, you will be tempted to deny points, which simply denotes location. If what you mean is that we cannot make an absolutely perfectly right angled body, that is both irrelevant and points to the real problem being the assumption that "real" means physical-material. That would show the pernicious influence of evolutionary materialistic scientism, which is self-refuting and necessarily false. But then, my saying that brings up propositions (the truth claims asserted or denied in statements) which strict materialists deny as being real. The problem then being that meaning, logical implication, logical antithesis and distinct identity also have not mass nor location nor size nor energy content and would become unreal, again illustrating the absurdities of such materialism and of fellow traveller movements. KF kairosfocus

Hazel@44, the same applies to right angles, straight lines and cubes. We can describe them mathematically, but they don’t really exist in nature. We can use math to model acceleration due to gravity, but at best we are talking about an average. Acceleration due to gravity on the bar stool I am currently sitting is different than it is on the stool at the other end of the bar. Ed George

Hazel, We can worry about the precise ontology and metaphysics of numbers, the continuum, motion involving rates of change tied to displacement, etc later. Right now, start from, is motion a real phenomenon? If not, explain how you typed your comments through the delusion of motion. If so, rates of change and connected locations in space across time are sufficient to underscore the point. KF kairosfocus

hazel, if you have a point, make it. I will not play your "no black swans" game. Is the Universe Made of Math?:

In this excerpt from his new book, Our Mathematical Universe, M.I.T. professor Max Tegmark explores the possibility that math does not just describe the universe, but makes the universe

ET

A circle is an abstract concept. No circle, in the mathematical sense, exists in the physical world. Do you agree, ET? hazel

hazel:

the thing I am unclear on is whether the abstract mathematics is “embedded” in the universe,

You are stuck in a box. Why do you think mathematics is only an abstraction? ET

Yes there is an "astonishing good fit" between mathematical descriptions and the physical world, and yes our world has structure, and yes distinct identity is the key foundation of mathematics. I don't think I disagree with any of that. If anything, and this may be a matter of semantics, or it may be, as I said earlier, a matter of philosophical point-of-view, the thing I am unclear on is whether the abstract mathematics is "embedded" in the universe, or whether those abstractions correctly model (with Einstein's caveat) a universe which has regularities in all of its multitudinous particular instances, but doesn't actually contain any abstract, Platonic overlay. hazel

Hazel and GE, we are not principally discussing particular laws or theories of science (though the exactitude of quantum results gives pause) but the quantities and structure of the world that renders it and its constituents substantially mathematical. That said, WJM is right to point out that mathematical description and frameworks have an astonishingly good fit that is also suggestive. We already saw that for any particular possible world W, it must manifest distinct identity so that W = {A|~A} which you cannot effectively deny on pain of finding indistinguishable things to be identical. Once that is present, nullity, unity and duality are, thence by successive order types the natural numbers thus also transfiniteness. From such the additive inverses, rationals and power series of ratios required for irrationals yield the continuum, which is manifest in space. Thence, displacement, its rate of change and its second rate give us trajectories of accelerated motion. All of this is before we can try to express a law. We see structure and quantity inextricably and pervasively embedded in the world. That is, the substance of mathematics. All of this has been pointed out, you have had little or no answer apart from verbal shifts and declaration of differences of opinion. That pattern speaks, and not in favour of the objections made. KF kairosfocus

Hazel@39. I like it. I think that his quote is an accurate reflection of our use of mathematics to model reality. It may work very well in many cases, but it is not perfect. As we would expect for any human endeavour. Ed George

A quote of Einstein's to ponder: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." hazel

Honestly, if anyone thought that finding a perfect sphere or cube would be a sign of intelligent design, how can they not see the perfectly mathematical behavior of phenomena as such? It's so bizarre that we think of objects that don't follow precise mathematical behaviors as being something in further need of explanation. It's like requiring further explanation for any rough spheroid shape of rock we find because it's NOT a perfect sphere. William J Murray

hazel, So then you can say mathematical properties exist in the real world, just as you can say "treeness" properties exist in the real world. It really seems, from this perspective, that some people are just trying to generate a semantic or conceptual dodge. The mathematical precision of behaviors in the real world is a real thing - as real and precise as if we found a perfect sphere in the real world. Continue the line of reasoning and terms you have started. It would be one thing if phenomena behaved in a "sort-of" mathematical way; but they do not. They behave with incredible mathematical precision. So, if you're going to commit to "circularity" or "treeness" being an actual quality (represented via symbolic language) that is actual and inherent in things in the universe, then even more so should you find "mathematics" an actual, inherent quality of the real world, because we're not just finding "sphereness" in these equations, so to speak - we've found perfect spheres (precise, perfectly mathematical behaviors). It's not just "order" we are looking at; it's wondrous order of incredible, mathematical precision. There's literally no other way to describe it. When you look at these behaviours, you might as well be looking at the physical manifestation of the mathematical equation. The equation was there for us to discover, we did not invent it. What we did was create a symbolic language with which to represent that equation. We did not invent the equation that acceleration is inversely proportional to mass given the same force; that mathematical equation existed, whether we expressed that equation in one language or another. We just discovered it. If mathematics wasn't sewn into the fabric of the universe as a discoverable feature (whether on purpose or not), why would that equation **always** hold true? Or any physics equation? We're not finding "mathematics-ness" like "sphere-ness"; the universe expresses perfect, precise mathematics. William J Murray

An analogy is a TV cooking show. We can say the cook is cracking an egg, mixing a cake, baking in the oven, etc. However, if we were to dissect the signals emanating from the screen, none of those elements described would be found in the signals. Do we then conclude the cook, egg, cake and oven do not exist? EricMH

Hazel:

But in what sense is “treeness” real? It’s not real in the same sense that a particulur tree is real. It’s real as an abstraction about the physical world, but it doesn’t seem to me that it exists in the physical world . . . . As I think about this, I understand, I think, that no one is claiming that abstract concepts themselves are part of the physical world. But other than the fact that the world is such that mathematical concepts can accurately describe and model or map the world, I’m not sure what else can be said. If the concepts aren’t in the physical world, what is there that could be called “math” that is in the physical world?

Here we come up against our struggle to move beyond a hidden premise of the materialistic age: real implies physical. The evidence is, that things can be that are not physical, and the evidence is that intelligence-bearing information, mind and mathematical frameworks affect and constrain or are even expressed in physical reality but are not themselves equal to signals or bodies. One hint is that quantum weirdness seems to be suspiciously similar. KF kairosfocus

As I think about this, I understand, I think, that no one is claiming that abstract concepts themselves are part of the physical world. But other than the fact that the world is such that mathematical concepts can accurately describe and model or map the world, I'm not sure what else can be said. If the concepts aren't in the physical world, what is there that could be called "math" that is in the physical world? I think, as Rob Sheldon alluded to above, this is a perennially slippery subject in which, in a Gestaltish way, one can slide back and forth between a Platonic and an Aristotelian view. Or even from a Platonic view, between seeing the Platonic ideals as mirror-like descriptions of the world as opposed to somehow causally informing the word. Interesting topic, but probably mostly a philosophical point-of-view issue without any way to settle the issue. hazel

Thanks. I understand that "treeness" is the set of properties that real trees have, and that there is a group of real objects, with real properties, that are properly classified as a "tree". But in what sense is "treeness" real? It's not real in the same sense that a particulur tree is real. It's real as an abstraction about the physical world, but it doesn't seem to me that it exists in the physical world. So when you write that

Does “circularity” exist? Yes, if by that we define it as “the properties that all circles have”

I agree. I just can't grasp that that is the same as saying circularity exists in the real world. The real physical world exists with a practically infinite amount of detail: it just is what it is. It is describable by math, and this is a wonderful fact about it, but that is different than saying the abstract concepts we use to describe it are actually part of the physical world. The individual properties characteristics of the world are physically real, but abstractions such as "the properties that all circles have (or trees)" are not physically real. That's the way it seems to me, I guess. hazel

Ed George gives primacy to what is 'real' to the physical universe, and holds that the mathematics discovered by the mind of man that, (thus far as far as our best scientific testing of the mathematical predictions will allow), perfectly describe the universe, are just an imaginary invention of man's mind, not a discovery. Yet, common sense (and quantum mechanics) gives primacy to what is 'real' to the mind of man, and even to the Mind of God, that 'invented' mathematics, not to the universe.

“The principal argument against materialism is not that illustrated in the last two sections: that it is incompatible with quantum theory. The principal argument is that thought processes and consciousness are the primary concepts, that our knowledge of the external world is the content of our consciousness and that the consciousness, therefore, cannot be denied. On the contrary, logically, the external world could be denied—though it is not very practical to do so. In the words of Niels Bohr, “The word consciousness, applied to ourselves as well as to others, is indispensable when dealing with the human situation.” In view of all this, one may well wonder how materialism, the doctrine that “life could be explained by sophisticated combinations of physical and chemical laws,” could so long be accepted by the majority of scientists." – Eugene Wigner, Remarks on the Mind-Body Question, pp 167-177. He goes toe-to-toe with science big wigs… and so far he’s undefeated. - interview Dr. Bernardo Kastrup: You see we always start from the fact that we are conscious. Consciousness is the only carrier of reality and existence that we can know. Everything else is abstraction; [they] are inferences we make from consciousness. http://www.skeptiko.com/274-bernardo-kastrup-why-our-culture-is-materialistic/ "In any philosophy of reality that is not ultimately self-defeating or internally contradictory, mind – unlabeled as anything else, matter or spiritual – must be primary. What is “matter” and what is “conceptual” and what is “spiritual” can only be organized from mind. Mind controls what is perceived, how it is perceived, and how those percepts are labeled and organized. Mind must be postulated as the unobserved observer, the uncaused cause simply to avoid a self-negating, self-conflicting worldview. It is the necessary postulate of all necessary postulates, because nothing else can come first. To say anything else comes first requires mind to consider and argue that case and then believe it to be true, demonstrating that without mind, you could not believe that mind is not primary in the first place." - William J. Murray “No, I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness.” Max Planck (1858–1947), the main founder of quantum theory, The Observer, London, January 25, 1931 “Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.” Schroedinger, Erwin. 1984. “General Scientific and Popular Papers,” in Collected Papers, Vol. 4. Vienna: Austrian Academy of Sciences. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden. p. 334.

Advancements in Quantum Mechanics, (a thoroughly mathematical theory with blatantly spiritual overtones), have now experimentally verified this 'common sense' conclusion that mind must be primary for anything else to be real. Specifically, advances in quantum mechanics have now falsified 'realism', which is the belief that the universe exists independently from mind, apart from any conscious observation.

Should Quantum Anomalies Make Us Rethink Reality? Inexplicable lab results may be telling us we’re on the cusp of a new scientific paradigm By Bernardo Kastrup on April 19, 2018 Excerpt: ,, according to the current paradigm, the properties of an object should exist and have definite values even when the object is not being observed: the moon should exist and have whatever weight, shape, size and color it has even when nobody is looking at it. Moreover, a mere act of observation should not change the values of these properties. Operationally, all this is captured in the notion of “non-contextuality”: ,,, since Alain Aspect’s seminal experiments in 1981–82, these predictions (of Quantum Mechanics) have been repeatedly confirmed, with potential experimental loopholes closed one by one. 1998 was a particularly fruitful year, with two remarkable experiments performed in Switzerland and Austria. In 2011 and 2015, new experiments again challenged non-contextuality. Commenting on this, physicist Anton Zeilinger has been quoted as saying that “there is no sense in assuming that what we do not measure [that is, observe] about a system has [an independent] reality.” Finally, Dutch researchers successfully performed a test closing all remaining potential loopholes, which was considered by Nature the “toughest test yet.”,,, It turns out, however, that some predictions of QM are incompatible with non-contextuality even for a large and important class of non-local theories. Experimental results reported in 2007 and 2010 have confirmed these predictions. To reconcile these results with the current paradigm would require a profoundly counterintuitive redefinition of what we call “objectivity.” And since contemporary culture has come to associate objectivity with reality itself, the science press felt compelled to report on this by pronouncing, “Quantum physics says goodbye to reality.” The tension between the anomalies and the current paradigm can only be tolerated by ignoring the anomalies. This has been possible so far because the anomalies are only observed in laboratories. Yet we know that they are there, for their existence has been confirmed beyond reasonable doubt. Therefore, when we believe that we see objects and events outside and independent of mind, we are wrong in at least some essential sense. A new paradigm is needed to accommodate and make sense of the anomalies; one wherein mind itself is understood to be the essence—cognitively but also physically—of what we perceive when we look at the world around ourselves. https://blogs.scientificamerican.com/observations/should-quantum-anomalies-make-us-rethink-reality/ An experimental test of non-local realism - 2007 Simon Gröblacher, Tomasz Paterek, Rainer Kaltenbaek, Caslav Brukner, Marek Zukowski, Markus Aspelmeyer & Anton Zeilinger Abstract: Most working scientists hold fast to the concept of ‘realism’—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned. http://www.nature.com/nature/journal/v446/n7138/full/nature05677.html Do we create the world just by looking at it? - 2008 Excerpt: In mid-2007 Fedrizzi found that the new realism model was violated by 80 orders of magnitude; the group was even more assured that quantum mechanics was correct. Leggett agrees with Zeilinger that realism is wrong in quantum mechanics, but when I asked him whether he now believes in the theory, he answered only “no” before demurring, “I’m in a small minority with that point of view and I wouldn’t stake my life on it.” For Leggett there are still enough loopholes to disbelieve. I asked him what could finally change his mind about quantum mechanics. Without hesitation, he said sending humans into space as detectors to test the theory.,,, (to which Anton Zeilinger responded) When I mentioned this to Prof. Zeilinger he said, “That will happen someday. There is no doubt in my mind. It is just a question of technology.” Alessandro Fedrizzi had already shown me a prototype of a realism experiment he is hoping to send up in a satellite. It’s a heavy, metallic slab the size of a dinner plate. http://seedmagazine.com/content/article/the_reality_tests/P3/ Experimental non-classicality of an indivisible quantum system - Zeilinger 2011 Excerpt: Page 491: "This represents a violation of (Leggett's) inequality (3) by more than 120 standard deviations, demonstrating that no joint probability distribution is capable of describing our results." The violation also excludes any non-contextual hidden-variable model. The result does, however, agree well with quantum mechanical predictions, as we will show now.,,, https://vcq.quantum.at/fileadmin/Publications/Experimental%20non-classicality%20of%20an%20indivisible.pdf Albert Einstein vs. Quantum Mechanics and His Own Mind – video https://www.youtube.com/watch?v=vxFFtZ301j4

in the following experiment, that was performed with atoms instead of photons, it was proved that measurement is everything. At the quantum level, reality does not exist if you are not looking at it,"

Experiment confirms quantum theory weirdness - May 27, 2015 Excerpt: The bizarre nature of reality as laid out by quantum theory has survived another test, with scientists performing a famous experiment and proving that reality does not exist until it is measured. Physicists at The Australian National University (ANU) have conducted John Wheeler's delayed-choice thought experiment, which involves a moving object that is given the choice to act like a particle or a wave. Wheeler's experiment then asks - at which point does the object decide? Common sense says the object is either wave-like or particle-like, independent of how we measure it. But quantum physics predicts that whether you observe wave like behavior (interference) or particle behavior (no interference) depends only on how it is actually measured at the end of its journey. This is exactly what the ANU team found. "It proves that measurement is everything. At the quantum level, reality does not exist if you are not looking at it," said Associate Professor Andrew Truscott from the ANU Research School of Physics and Engineering. Despite the apparent weirdness, the results confirm the validity of quantum theory, which,, has enabled the development of many technologies such as LEDs, lasers and computer chips. The ANU team not only succeeded in building the experiment, which seemed nearly impossible when it was proposed in 1978, but reversed Wheeler's original concept of light beams being bounced by mirrors, and instead used atoms scattered by laser light. "Quantum physics' predictions about interference seem odd enough when applied to light, which seems more like a wave, but to have done the experiment with atoms, which are complicated things that have mass and interact with electric fields and so on, adds to the weirdness," said Roman Khakimov, PhD student at the Research School of Physics and Engineering. http://phys.org/news/2015-05-quantum-theory-weirdness.html

The Theistic implications of this experiment are fairly obvious. As Professor Scott Aaronson quipped, “Look, we all have fun ridiculing the creationists,,, But if we accept the usual picture of quantum mechanics, then in a certain sense the situation is far worse: the world (as you experience it) might as well not have existed 10^-43 seconds ago!”

“Look, we all have fun ridiculing the creationists who think the world sprang into existence on October 23, 4004 BC at 9AM (presumably Babylonian time), with the fossils already in the ground, light from distant stars heading toward us, etc. But if we accept the usual picture of quantum mechanics, then in a certain sense the situation is far worse: the world (as you experience it) might as well not have existed 10^-43 seconds ago!” – Scott Aaronson – MIT associate Professor quantum computation - Lecture 11: Decoherence and Hidden Variables

Of related note:

BRUCE GORDON: Hawking’s irrational arguments – October 2010 Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy. This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,, Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,, Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/

Verse:

Colossians 1:17 He is before all things, and in him all things hold together.

bornagain77

Hazel, not only does treeness exist but key properties involved are informational, encoded in a language, in dna. Codes, as we know, are massively structured and quantitative phenomena, so are mathematical.KF kairosfocus

hazel: Trees are real in the world, but is “treeness”? Yes. "Treeness" is the real set of properties that real trees have. What you're really talking about is classification. Another way to refer to "treeness" is that there is a class of real objects we call "tree" that have a certain set of real properties. But "treeness" is certainly a valid shorthand (and folksy) way of describing this set of properties that trees have. Does "circularity" exist? Yes, if by that we define it as "the properties that all circles have." mike1962

I agree in part with what wjm says, but there is one distinction I am unclear about. Mathematics uses symbolic representations to describe real things. The particular systems of symbolic representations we use are human inventions. For instance, the famous difference between the notations for derivatives offered by Newton and Leibnitz, where we finally settled on Leibnitz's. Also, the world obviously has structure, order, and regularities which can be described mathematically. I really like the quote in the OP: ”Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision." Here is the part I am unclear on. wjm says,

The symbols contained in mathematics are symbolic representations of those real things – like the word “water” is symbolic of the real thing, or the term “tree” is a symbolic representation of the real thing. It makes no more sense to say that humans invented mathematics than it makes sense to say that humans invented water or trees.

There are real specific trees. However the word "tree" is an abstract concept that does not represent any specific real tree, but applies to a set of general characteristics which are present in specific trees. So in whatever sense the concept represented by the word "tree" is real, it is real in a different way than actual trees are real. We didn't invent trees. We did invent the word "tree". Trees are real in the world, but is "treeness"? Applying these thoughts to math (and I'm sure the situation is different, but I'm not sure how), spherical things exists, and the concept of sphere can represent a certain common property (all points equidistant from a center point). However, real spheres are never perfectly spherical, and they all have properties other than their sphericalness. So in what sense is the mathematical concept of sphere actually in the real world? hazel

Did Fibonacci invent the Fibonacci sequence? The Fibonacci sequence is “a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding natural numbers”: 0+1=1;1+1=2; 1+2=3; 2+3=5; 3+5=8; 5+8=13; 8+13=21… (forever) 1, 2,3,5,8,13,21… are the Fibonacci numbers. Some really interesting things happen when you apply this sequence to geometry by tiling the numbers: https://en.wikipedia.org/wiki/Fibonacci_number#/media/File:FibonacciSpiral.svg This seems to be nothing more than a clever parlor trick, so it seems like something that Fibonacci invented.* Right? But if that’s true why do we find the Fibonacci sequence in nature? Take a look at the first ten minutes of this episode of Nova, which is a regular science program on PBS. The episode gives several examples of where the Fibonacci sequence as well as the value of pi appear, sometime quite unexpectedly, in nature. It’s followed by an interview with MIT physicist Max Tegmark who believes that everything in nature can be reduced to mathematics. He appears to suggest that we could all be living in some sort of virtual reality. https://www.youtube.com/watch?v=mpcpzXuzdQk The whole documentary is worth watching because it is examining the very same question that we are considering here. *Actually Indian mathematicians "discovered" the sequence about 1200 years earlier. john_a_designer

kf@25, Well said. What I always find most interesting is the cognitive dissonance on display. Here we have someone who agrees that what mathematics is describing are real, natural aspects of the world, but will not agree that mathematics is a real, natural aspect of the world. Yes, we all know the word itself, the symbolic representation itself, the model itself is not the reality, but that's true of every word or phrase or model we use to represent the reality. To say "humans invented mathematics" is therefore the precise equivalent to saying "humans invented inertia" or "humans invented horses." I mean, there are a lot of things that humans invented, but the mathematical behavior of aspects of the universe, and as a foundational aspect of identity and cognitive thought, is not among those inventions. Some of those discoverable aspects of existence and the universe are so ubiquitous, like identity and gravity, it takes a very keen mind to even notice them and a very stubborn mind to ignore or deny once revealed. William J Murray

hazel said:

There is lots of math that doesn’t model anything real. Does that make it useless? Or have I missed something in the preceding conversation about this?

Any language can be used to describe both real and unreal things. That's a completely irrelevant point. Nobody is trying to claim that the symbolic representations themselves are real in the sense that they are discoverable - we didn't go out and find "2+3=5" in a natural rock formation, just as we didn't find the English world "rock" or "tree" in the rock or in the tree. Does inertia exist in the real world? Does gravity? Does electromagnetic radiation? Does the way phenomena behave in relationship to those things actually exist in the real world? We discovered those things and the related behaviors. We applied symbolic language to describe them. It makes no more sense to say "we invented mathematics" than it makes sense to say "we invented inertia, gravity and electromagnetic radiation." It's like saying we invented the fact that if you add two apples to three apples you have five apples. No, we did not - we started using a specific language to represent an experiential fact. As with all language terms that represent real things, mathematics represents a real thing we find in the universe. ALL such words and terms are representational. William J Murray

WJM, I am inclined to hold that in our studies we do bring to bear cultural features, but the focal substance is structure and quantity. So, reality has a mathematical aspect and we may investigate it as a project of our civilisation. The discipline is not the substance, and the discipline recognises the substance. Indeed, even as "horse" recognises certain noble animals, "one," or 1 or I etc recognise the reality of unity. Similarly "two" etc recognise that there is duality, inevitably in any distinct possible world. Our challenge, I believe, is that the Kantian ugly gulch has now pervaded our culture and we think we are locked up in a world of appearances, shared memes etc that cannot bridge to things in themselves. That is, we have discredited truth and are beginning to fall into chaos. The loss of ability to recognise the manifest reality of say maleness and femaleness leading to 112 "genders" at last count, is but one symptom of that chaos. Far more central is what we are seeing, radical relativisation and cultural captivity of science and mathematics that locks them away from seeking and being motivated and governed by truth, accurate description of reality. The question haunts me: has our culture gone suicidally insane, caught up in Plato's Cave shadow-shows? KF kairosfocus

ES @ 10: "objective and complete knowledge of the world"? That seems to be perilously close to omniscience! Did you ever find any scientists who were that inclined to play at being God? I would think that seeking observationally reliable, substantial and useful understanding and knowledge of the world tempered by due awareness of our epistemic limitations would be enough for creatures such as we are. At the same time, I find it highly significant that knowledge requires acknowledgement of what is well-warranted, i.e. belief, that seemingly bitter pill to swallow. As in, well-warranted (so, reliable), credibly true belief. Without truth-seeking and humility before truth, however, the moral mainspring of knowledge is broken. So, I am disinclined to reduce science and mathematics to modelling, which explicitly surrenders truth-seeking in interests of utility and simplicity. We may use models in such disciplines and we must be aware that error exists but truth -- accurate description of reality -- must ever be a goal and chief virtue. KF kairosfocus

Ed George said:

I don’t get your point. Why would we attempt to model something that isn’t real?

Mr. George, do you not understand that the term "mathematics", and the formulas that describe the behavior of phenomena, are symbolic representations of something you agree is REAL? You admit that what mathematics represents is real. The symbols contained in mathematics are symbolic representations of those real things - like the word "water" is symbolic of the real thing, or the term "tree" is a symbolic representation of the real thing. It makes no more sense to say that humans invented mathematics than it makes sense to say that humans invented water or trees. William J Murray

EG @ 4: The issue isn't conclusion jumping but conclusion resisting! There is no question but that the world has in it features or aspects that are quantitative or structural, which is the substance of mathematics. Further, distinct identity of any possible world directly entails duality, unity, nullity thence the natural numbers, i.e. countable quantity. So, it's not a happenstance of our world, it is framework for any world. The existence of a world also inevitably requires order instead of utter chaos, i.e. structure. Again, the substance of mathematics. Also, that distinct stable identity of a world implies that events, circumstances and objects within it will reflect core ordering characteristics so we can ground inductive exploration and discovery of such rational principles, aka laws; despite our error-prone epistemic limitations, we are not reduced to despair that we can never cross the ugly gulch to understand (in part at least) things as they are. That is empirical, reliable, well-warranted and even accurate -- truthful -- knowledge of the observed cosmos is possible. Further, we have in fact found a framework of such laws, which are intensely mathematical, and exhibit fine tuning that sets our world at a deeply isolated operating point in the space of mathematically possible frameworks. And yes, that strongly points to design of the cosmos by an intelligent agent whose mind is also inclined to mathematics; though that is not our primary focus here. KF kairosfocus

ES, yes our observations are error prone but they also often hit the mark: e.g. there are exactly four letters in m-1 a-2 r-3 k-4; this is of course a quantitative phenomenon. Likewise, rational reflection and insights reveal self-evident first truths and principles. Identity being central, and opening up how quantity and structure (the substance as opposed to the study of Mathematics) are inescapably integral to the framework for any possible world. KF kairosfocus

H, the issue is not all of Mathematics or of its extensions into abstract logic model worlds, but a key core. That embedding starts with distinct identity and its correlate, distinction thus duality, unity, nullity. From this the von Neumann succession of order types directly brings out the naturals. Which, lead to the integers, the rationals, the reals and many properties and manifestations. For example falling implies continually changed location thus length and the continuum. It also brings in infinitesimals and/or limits (thus, power series) as we address rates dx/dt --> v, dv/dt --> a, F = dP/dt, thus F = ma when m is constant. We cannot even properly observe and measure the phenomena without recognising the centrality of structure and quantity. Which, is the substance (as opposed to the study) of Mathematics. Going further, there is a tendency to trivialise the pervasiveness of pi and e in all sorts of phenomena, and the way e, pi, i, 0 and 1 are dovetailed to infinite precision in the Euler expression. Properties such as primeness and irrationality (e.g. sqrt-2, which popped up in pondering the ratio of sides to diagonal of a square) come out. While we are on the Pythagoreans, they noticed how we instinctively respond to whole number ratios in music, indeed the ratios were found at the root of pleasing musical patterns -- transforming how we understand aesthetics. There is much more, all pointing to how quantity and structure are embedded in our cosmos and indeed any possible world. Which on massive evidence invites study and is integral to the intelligibility of the world. Which in turn is why we find quantitative laws, theories and models so useful. All of which then highlights the need for a more insightful and more accurate understanding. KF kairosfocus

There is lots of math that doesn't model anything real. Does that make it useless? Or have I missed something in the preceding conversation about this? I see, I think, after looking back a few points. The issue is whether the math is actually real in the world, as opposed to in our symbolic system. The argument seems to be that if the model fits the real world so well, as it does, then it makes sense to say the math is in the world and not just i the model. Do I understand your point, Mike? hazel

Mike1962

What other way is there to look at it? The mathematics are useless unless the thing it actually models is real. (And does so with astonishing precision.)

I don’t get your point. Why would we attempt to model something that isn’t real? Ed George

Euler's identity is neat, and is one of many marvelous math facts, but I don't think Euler's identity is embedded in the universe anyplace, nor models any specific phenomena. I might be wrong, though, and would welcome being corrected by an example. hazel

Eugene S: Any such description is bound to be limited in accuracy and to include some cognitive bias. As soon as there is an epistemic cut between the observer and the observed, some subjectivity is there. Therefore scientific knowledge has the disturbing property of never being able to deliver complete knowledge. I can agree with that, however, e, Pi, and i appear over and over again in our best and extremely accurate models and physical descriptions, and are curiously related as described by Euler's Identity. How is that true if those values, and that relation, are not baked into the universe in a fundamental way? mike1962

Ed George: ...to jump to the conclusion that, because mathematics is good at modelling the universe, that mathematics is inherent in the universe is not warranted. What other way is there to look at it? The mathematics are useless unless the thing it actually models is real. (And does so with astonishing precision.) Pi shows up everywhere. E shows up everywhere. i shows up everywhere. Quantum physics, the most precise set of descriptions and means of prediction the world has seen, depends on the use of all of these. As does General Relativity and a hosts of other physical phenomena. And before quantum physics came along, Euler proved the cozy relation of e^ i pi + 1 = 0. Whoa Nellie! So if these mathematical facts are not baked into the universe, why does the universe act like they are? With astonishing precision. It is possible that our model is not completely accurate, but so far there is no indication of it. And even if were are somewhat wrong, something very close to what the mathematics describes would have to be true... long before humans invented mathematics. mike1962

Moreover, the anomalies that are now found in the “Cosmic Microwave Background Radiation” (CMBR) (which the perfect flatness of the universe allows us to study in detail), highlights just how unique our solar system and earth are in this universe,,

In other words, the “tiny temperature variations” in the CMBR, (from the large scale structures in the universe, to the earth and solar system themselves), reveal teleology, (i.e. a goal directed purpose, a plan), that specifically included the earth from the start. ,,, The earth, from what our best science can now tell us, is not a random cosmic fluke as atheists presuppose. https://uncommondesc.wpengine.com/intelligent-design/our-solar-system-is-a-lot-rarer-than-it-was-a-quarter-century-ago/#comment-669546

Besides the flatness of the universe, "Platonic perfection" is also found in other places in physics. One little known 'platonic perfection' is the higher dimensional "amplituhedron" of Quantum-electrodynamics that was found a few years back:

Bohemian Gravity - Rob Sheldon - September 19, 2013 Excerpt: Quanta magazine carried an article about a hypergeometric object that is as much better than Feynman diagrams as Feynman was better than Heisenberg's S-matrices. But the discoverers are candid about it, "The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity. “Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”" What are these suspect principles? None other than two of the founding principles of materialism--that there do not exist "spooky-action-at-a-distance" forces, and that material causes are the only ones in the universe.,,, http://rbsp.info/PROCRUSTES/bohemian-gravity/

And the other place(s) where 'platonic perfection' is found is in the 4-Dimensional space time curvature of special and general relativity. Simply put, no experimental test to date has ever been able to detect any 'imperfection' for what the theory(s) of relativity predict (for higher dimensional 4-D space-time curvature and/or geometry). As Berlinski noted,

“On the other hand, I disagree that Darwin’s theory is as `solid as any explanation in science.; Disagree? I regard the claim as preposterous. Quantum electrodynamics is accurate to thirteen or so decimal places; so, too, general relativity. A leaf trembling in the wrong way would suffice to shatter either theory. What can Darwinian theory offer in comparison?” (Berlinski, D., “A Scientific Scandal?: David Berlinski & Critics,” Commentary, July 8, 2003)

Supplemental note:

Quantum Mechanics, Special Relativity, General Relativity and Christianity - video https://www.youtube.com/watch?v=h4QDy1Soolo

bornagain77

EugeneS mentions “imperfections” in spheres, (and supposedly also imperfections in triangles, squares, lines, etc.. etc..), and indeed there are no perfect spheres (nor any other perfect Euclidean objects in this universe). It is also interesting to note the history that assuming perfection has had in science. Copernicus, (who was heavily influenced by Platonic thinking), imagined (incorrectly) that the planets move in perfect circles (rather than ellipses). Later, Newton, for allowing God could adjust the orbits of the planets, was chastised by Leibniz, (and Laplace) for having a "very narrow ideas about the wisdom and the power of God.".. i.e. For having a narrow view of the perfection of God. Laplace, contrary to atheistic folklore, cited with approval Leibniz's criticism of Newton's invocation of divine intervention to restore order to the Solar System: "This is to have very narrow ideas about the wisdom and the power of God.", to them, it would count as evidence against intelligent design if God had to intervene to prevent the solar system from collapsing. So intelligent design could just as easily be a motivation to prove the stability of the solar system.

"Leibniz, in his controversy with Newton on the discovery of infinitesimal calculus, sharply criticized the theory of Divine intervention as a corrective of the disturbances of the solar system. "To suppose anything of the kind", he said, "is to exhibit very narrow ideas of the wisdom and power of God'." - Pierre-Simon Laplace https://books.google.com/books?id=oLtHAAAAIAAJ&pg=PA73&lpg=PA73

Moreover, contrary to what is commonly believed, Laplace did not really solve the problem of planetary perturbations in the end, (he only solved for for first degree approximations), but Haret showed that orbits are not absolutely stable using third degree approximations. Moreover, I hold that if Newton and Leibniz (and even Laplace) could see our science today they would be very pleased by what modern science has now revealed about the wisdom and power of God:

“You might also think that these disparate bodies are scattered across the solar system without rhyme or reason. But move any piece of the solar system today, or try to add anything more, and the whole construction would be thrown fatally out of kilter. So how exactly did this delicate architecture come to be?” R. Webb - Unknown solar system 1: How was the solar system built? - New Scientist – 2009 Is the Solar System Stable? By Scott Tremaine - 2011 Excerpt: So what are the results? Most of the calculations agree that eight billion years from now, just before the Sun swallows the inner planets and incinerates the outer ones, all of the planets will still be in orbits very similar to their present ones. In this limited sense, the solar system is stable. However, a closer look at the orbit histories reveals that the story is more nuanced. After a few tens of millions of years, calculations using slightly different parameters (e.g., different planetary masses or initial positions within the small ranges allowed by current observations) or different numerical algorithms begin to diverge at an alarming rate. More precisely, the growth of small differences changes from linear to exponential:,,, As an example, shifting your pencil from one side of your desk to the other today could change the gravitational forces on Jupiter enough to shift its position from one side of the Sun to the other a billion years from now. The unpredictability of the solar system over very long times is of course ironic since this was the prototypical system that inspired Laplacian determinism. Fortunately, most of this unpredictability is in the orbital phases of the planets, not the shapes and sizes of their orbits, so the chaotic nature of the solar system does not normally lead to collisions between planets. However, the presence of chaos implies that we can only study the long-term fate of the solar system in a statistical sense, by launching in our computers an armada of solar systems with slightly different parameters at the present time—typically, each planet is shifted by a random amount of about a millimeter—and following their evolution. When this is done, it turns out that in about 1 percent of these systems, Mercury’s orbit becomes sufficiently eccentric so that it collides with Venus before the death of the Sun. Thus, the answer to the question of the stability of the solar system—more precisely, will all the planets survive until the death of the Sun—is neither “yes” nor “no” but “yes, with 99 percent probability.” https://www.ias.edu/about/publications/ias-letter/articles/2011-summer/solar-system-tremaine

Along this line of "perfection" thought, it is interesting to note where in this universe perfection for spheres is approached rather closely,,,

Sun's Almost Perfectly Round Shape Baffles Scientists - (Aug. 16, 2012) — Excerpt: The sun is nearly the roundest object ever measured. If scaled to the size of a beach ball, it would be so round that the difference between the widest and narrow diameters would be much less than the width of a human hair.,,, They also found that the solar flattening is remarkably constant over time and too small to agree with that predicted from its surface rotation. http://www.sciencedaily.com/releases/2012/08/120816150801.htm Bucky Balls - Andy Gion Excerpt: Buckyballs (C60; Carbon 60) are the roundest and most symmetrical large molecule known to man. Buckministerfullerine continues to astonish with one amazing property after another. C60 is the third major form of pure carbon; graphite and diamond are the other two. Buckyballs were discovered in 1985,,, http://www.3rd1000.com/bucky/bucky.htm

The delicate balance at which carbon is synthesized in stars is truly a work of art.,,, Years after Sir Fred discovered the stunning precision with which carbon is synthesized in stars he stated this:

"I do not believe that any physicist who examined the evidence could fail to draw the inference that the laws of nuclear physics have been deliberately designed with regard to the consequences they produce within stars." Sir Fred Hoyle - "The Universe: Past and Present Reflections." Engineering and Science, November, 1981. pp. 8–12

And perfection for a sphere is also approached in the Cosmic Background Radiation (CBR). ,,, Of the supposed "imperfections" in the sphere of the CBR, the following author comments, "the discovery of small deviations from smoothness (anisotopies) in the cosmic microwave background is welcome, for it provides at least the possibility for the seeds around which structure formed in the later Universe"

The Cosmic Background Radiation Excerpt: These fluctuations are extremely small, representing deviations from the average of only about 1/100,000 of the average temperature of the observed background radiation. The highly isotropic nature of the cosmic background radiation indicates that the early stages of the Universe were almost completely uniform. This raises two problems for (a naturalistic understanding of) the big bang theory. First, when we look at the microwave background coming from widely separated parts of the sky it can be shown that these regions are too separated to have been able to communicate with each other even with signals traveling at light velocity. Thus, how did they know to have almost exactly the same temperature? This general problem is called the horizon problem. Second, the present Universe is homogenous and isotropic, but only on very large scales. For scales the size of superclusters and smaller the luminous matter in the universe is quite lumpy, as illustrated in the following figure. ,,, Thus, the discovery of small deviations from smoothness (anisotopies) in the cosmic microwave background is welcome, for it provides at least the possibility for the seeds around which structure formed in the later Universe. However, as we shall see, we are still far from a quantitative understanding of how this came to be. http://csep10.phys.utk.edu/astr162/lect/cosmology/cbr.html

It is also interesting to note where "platonic perfection" is, not only approached, but arguably reached in the universe.,,, The universe is perfectly flat as far as our best scientific instruments can tell us:

How do we know the universe is flat? Discovering the topology of the universe - by Fraser Cain - June 7, 2017 Excerpt: With the most sensitive space-based telescopes they have available, astronomers are able to detect tiny variations in the temperature of this background radiation. And here's the part that blows my mind every time I think about it. These tiny temperature variations correspond to the largest scale structures of the observable universe. A region that was a fraction of a degree warmer become a vast galaxy cluster, hundreds of millions of light-years across. The cosmic microwave background radiation just gives and gives, and when it comes to figuring out the topology of the universe, it has the answer we need. If the universe was curved in any way, these temperature variations would appear distorted compared to the actual size that we see these structures today. But they're not. To best of its ability, ESA's Planck space telescope, can't detect any distortion at all. The universe is flat.,,, We say that the universe is flat, and this means that parallel lines will always remain parallel. 90-degree turns behave as true 90-degree turns, and everything makes sense.,,, Since the universe is flat now, it must have been flat in the past, when the universe was an incredibly dense singularity. And for it to maintain this level of flatness over 13.8 billion years of expansion, in kind of amazing. In fact, astronomers estimate that the universe must have been flat to 1 part within 1×10^57 parts. Which seems like an insane coincidence. https://phys.org/news/2017-06-universe-flat-topology.html

Interestingly, this 'perfect flatness' is essential for us to be able to practice math and science,

Why We Need Cosmic Inflation By Paul Sutter, Astrophysicist | October 22, 2018 Excerpt: As best as we can measure, the geometry of our universe appears to be perfectly, totally, ever-so-boringly flat. On large, cosmic scales, parallel lines stay parallel forever, interior angles of triangles add up to 180 degrees, and so on. All the rules of Euclidean geometry that you learned in high school apply. But there’s no reason for our universe to be flat. At large scales it could’ve had any old curvature it wanted. Our cosmos could’ve been shaped like a giant, multidimensional beach ball, or a horse-riding saddle. But, no, it picked flat. https://www.space.com/42202-why-we-need-cosmic-inflation.html

bornagain77

Robert Sheldon,

It’s an old debate. “Do we discover math, or invent it?” Plato says we discover it. Aristotle says we invent it. Most mathematicians who have advanced beyond algebra, would be Platonists. Even those who claim to be Aristotelians will admit they are functionally Platonists.

According to mathematician Roger Penrose, who collaborated with Stephen Hawking on some of his early work, “mathematics seems to have its own kind of existence.” He then goes on to explain:

It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures… It also has this relationship to mentality because we certainly have access to mathematical truths. I think it is useful to think of the world as not being a creation of our minds because if we do then how could it have been there before we were around? If the world is obeying mathematical laws with extraordinary precision since the beginning of time, well, there were no human beings and no conscious beings of any kind around then. So how can mathematics have been the creation of minds and still been there controlling the universe? I think it is very valuable to think of this Platonic mathematical world as having its own existence. So let’s allow that and say that there are three different kinds of existence. There may be others, but three kinds of existence: the normal, physical existence; the mental existence (which seems to have, in some sense, an even greater reality – it is what we are directly aware of or directly perceive); and the mathematical world which seems to be out there in some sense conjuring itself into existence – it has to be there in some sense.

https://www.reasonablefaith.org/media/reasonable-faith-podcast/roger-penrose-interview-part-1/#_ftn3 Just to clarify, earlier in the interview Penrose described his metaphysical world view as a tripartite one consisting of the physical world, the mental world and a separate and distinct mathematical world. He goes on to explain that… ’there is the relationship between these three worlds which I regard, all three of them, as somewhat mysterious or very mysterious. I sometimes refer to this as “three worlds and three mysteries.” Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision. That’s mystery number one.’ john_a_designer

Good points by EugeneS at 9 and 10, if we accept not that we have given up the pursuit of truth, but rather that that pursuit can lead to "objective and complete knowledge of the world." hazel

"we are only describing them using symbolic language." Any such description is bound to be limited in accuracy and to include some cognitive bias. As soon as there is an epistemic cut between the observer and the observed, some subjectivity is there. Therefore scientific knowledge has the disturbing property of never being able to deliver complete knowledge. Post-Goedelian science has given up on its pursuit of the truth, i.e objective and complete knowledge of the world. The science of today is model-building, a humbler undertaking. EugeneS

"sphere... the form of an object in the real world" The problem is there is no such thing as a sphere in the real world. A sphere is an abstraction, a mathematical idea that focuses on a particular set of properties and disregards some "imperfections" inherent in any real thing. EugeneS

RS, indeed there has been a longstanding debate. Notwithstanding, the principle of distinct identity and understanding Mathematics as the [study of the] logic -- rational principles -- of structure and quantity gives a way forward. Once a distinct world W is, identity obtains, so we may freely write W = {A|~A}, thus we identify nullity, unity and duality. This sets up the succession of naturals per order type increment following von Neumann. Such helps us understand that some aspects of structure and quantity are framework to any possible world. Such are necessary entities embedded in the order of any possible world. Now too, in studying Mathematics we may set up abstract logic model worlds freely using symbols, axioms etc which would be possible worlds. In so doing we may encounter entities that are also necessary, quantitative and structural. These, we may detach and freely extend everywhere as they will be framework for any world. Indeed, we started with a case, the naturals. However, our creativity and cultural tradition are involved in how we set up models and in how we symbolise. Of course, several approaches may be substantially equivalent or at least sufficiently effective, e.g. classically the diverse formulations of Calculus. KF kairosfocus

Language and math came a lot easier to us because the first person Adam who was created in full stature on day 6 had both software programs installed. Of course we lost a lot of info along the way that we are still recovering :) reference: RCCF Framework for understanding science. Edenics.org on language, T' AZ 8a and chazal thereon, on the science/math.. Pearlman

It's an old debate. "Do we discover math, or invent it?" Plato says we discover it. Aristotle says we invent it. Most mathematicians who have advanced beyond algebra, would be Platonists. Even those who claim to be Aristotelians will admit they are functionally Platonists. Weirdly enough, the opposite is true in linguistics: more Aristotelians than Platonists. Perhaps because few people do algebra for fun, but everybody loves to talk. So most people have opinions about linguistics, and one of the most common opinions is that making up words, calling people names, inventing neologisms is so easy it needs no explanation. Rather, humans who can't talk need to have a very good explanation. But linguists who have moved beyond latin grammar and syntax, often are Platonists. They discover, like Noam Chomsky, that the inbred grammar of very small children needs an explanation. The limited syntax rules in all the world's populations begs for a cause. And indeed, the very paradigm for Charles Darwin's biological theorizing--the evolution of language--turns out to be a old wive's tale. Rather, language is a gift. And the gift is traceable. Robert Sheldon

Ed George- The universe was intelligently designed. Do you really think it was intelligently designed without the use of mathematics? If so then there isn't hope for you as a rational thinking person. Doug Adams choked. Any puddle that could contemplate its existence would also know that it is shrinking. ET

However, humans did not invent those behaviors

Agreed, but we did invent the means to model them, mathematics. But to jump to the conclusion that, because mathematics is good at modelling the universe, that mathematics is inherent in the universe is not warranted. That is like saying that because humans thrive on earth that the universe must be finely tuned for our existance. Douglas Adams had a humorous take on this in one of his books. Although there is plenty of randomness in the universe, it is also highly ordered (or non-random). Whenever anything has order, we can model it using mathematics. Does that mean that the mathematics preceded this order? Other than ET, I don't think that anyone is suggesting this. Ed George

I think they seek to work with mobile phones as a new dominant platform. I am struggling to see how to indent, esp when that is after the fact, and in some cases to insert diagrams from an existing archive. At least we can insert a vid now at UD . . . kairosfocus

I just found it overly complicated, but once I figured it out, it's not so bad. That's the problem with designers, they think more = better. I don't for the life of me see how the new interface is any "better," functionally, than the old. William J Murray

WJM, I didn't notice your response. KF PS: Do you find the new posting system as annoyingly cramped as I do? kairosfocus